How does an incoming commander deal with a weak 1SG?

Don't come into the Company with a preconceived notion about what is or isn't the problem. Take what the outgoing guy is saying with a grain of salt. He's on his way out, maybe he's had a rough time with the 1SG. But, the fact that he's leaving you with what he considers a unit that's "in the toilet", when HE is ultimately responsible for that as the Commander... well... consider the source.

Have an interview with the 1SG - one-on-one - not to find out what's wrong with him, but to get a candid view from his side of the story. When you get there, do a Command Climate survey - or, better yet, see if the outgoing Commander will do one on his way out.

DO NOT GO INTO THIS COMMAND THINKING YOU HAVE A PROBLEM TO FIX. Assess the situation once you are in the seat. Talk to your NCOs. Talk to your Soldiers. Let the outgoing Commander go in peace and look forward, not backward. You can't change what's done, but your attitude will set the tone going forward. Be positive, build your company, build your team and show your Soldiers you care about them first.

Cpt D K - You have your work cut out for you and it likely will be neither easy or pleasant.
You do not say in what areas he is weak or if he is simply not a leader or has no desire to lead.
Therefore I suggest you give him several simple tasks and one quite complicated and watch his behavior measuring his performance against the current leadership model to determine where he misses the mark on those points. This can also be used to either validate the opinion of your predecessor or debunk it. (Sometimes an SM will not perform well if the senior leader is toxic - although that should never be an issue with a first shirt)
Bring your training officer / XO into the conversation to learn what they have observed and when you have witnessed and heard enough, bring the matter to the SGM to get their opinion on how to best approach the problem so as to bring your FSG into compliance for retention purposes.
Somethings are considered "Sergeants Business" and are better left to the NCOs to sort out and the SGM may well tell you "Leave it to me, sir - I"ll handle it." I've known of such cases and it works reasonably well. Frankly speaking, I'm wondering where the SGM has been on this issue - perhaps he was told not to get involved and that is something to consider as well.
At any rate, it may come down to replacing him and letting the Army take care of the matter for you.
There is a great deal to consider here and I hope this will help you in one way or another.
Good luck with this.

CPT (Join to see) I would pursue this route:
- sit down with the CSM and have a fact finding conversation about your 1SG and what has been going on. Do this before you take over if at all possible. Three things: if you have to relieve the 1SG, you'll need the CSM on your side. If he isn't, your going To have to work with who you have. second: there may be something you are not being told that is key to your situational understanding. Finally, if you engage with the CSM in a developmental/collaborative way, you'll likely pick up an ally and cross off your list someone you'll have to battle. It may also set the stage for the flow of the right NCOs to your formation.
- you need a long term operational goal to focus on to drive what follows. This should come from mission or mutually with the BN CDR.
- set clear, easy to understand expectations of soldiers and NCOs in your command philosophy and initial counseling. Aim for the elegance of simplicity in straight forward easy to remember things YOU believe, not that sounded good in someone else's philosophy. Do this off the bat. Brief your 1SG one on one, explain you need him on board. Then brief your officers and your 1SG off to the side. Fuse your relationship with the 1SG. How much of your 1SGs weakness is tied to crap support and being written off at the jump? If he fails, at least you gave him the benefit of the doubt. Then just brief the NCOs. Then you do a Company level roll out of these things. By doing this, you let the NCOs and Officers be in an environment to ask questions and get clarification without the rank and file there. It sets conditions so when Joe starts asking questions of the NCOs, you are all one voice.
- issue orders and intent. Give it to Officers to see through, but make it clear what things you expect NCOs to supervise and execute. Build their confidence and ability in small things and turn up the intensity, accountability, and feedback as you go. You don't walk into the gym for the first time and bench 450. You have to build leadership muscles.
- the immediate way to build credibility for NCOs is to show they care. They have to build soldiers and help them solve problems. A soldier problem that they help solve is the single best opportunity to build that trust and credibility. Opportunity for your First Sergeant is to build a NCOPD program that works on skills, knowledge, and abilities of getting soldiers to the right resources to enable them to solve their problems, and follow up, and follow up, and follow up.
- the next biggest source of NCO credibility is making them technically/tactically proficient. Your First Sergeant should be certifying them on soldier skillls and ensuring they are squared away. Checking that they know how to train soldiers in the same. Once you have a solid base of 10 level skills, start getting the NCOs proficiency at Squad, then Platoon Level. At the same time your Officers must be doing the same. Sometimes just as Officers, sometimes as a team building with their senior NCOs.
- you'll have to figure out when you start collective training and exercises. Tough and realistic training is a huge morale builder. Follow it up with disciplined AARs and recovery.
- look at taskings as an opportunity to build mission competence, planning capacity, problem solving, and NCO expertise and confidence. Give a young Sergeant a mission and some resources and have them Plan and execute. Do the same with your LTs with additional duties and larger taskings.
- Pair and officer and an NCO for all additional duty areas. Same reasoning. Even the grind of a CIP can build capability and capacity while ensuring readiness.
- focus on the METL and readiness. The BS before, was the BS before.
- what you want to build is a reputation for your company being competent, exercising disciplined initiative, and generally squared away. When your soldiers see they are more squared away than others, they'll start believing it, and morale will improve. Keep throwing challenges right on the edge of their capability so they can do it, but it will be a stretch.
- seriously get all your systems and processes to a B plus. Delegate as much as you can and spend time looking forward.
- a lot of success in company command grows directly from taking care of soldiers, building NCOs and officers worthy of being stolen by BN, doing routine stuff routinely, and being able to actually do what you say your company can do.
- unless you follow up and check on things you'll get yep'ed and nothing will change. Hold leader's feet to the fire. Follow up, schedule it on your calendar.
- Soldiers need to see you as someone who cares, but is tough, competent, practical, and takes no shit. But you have to be your authentic self. You can't play a part for 12-18 months.