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Assuming that is it not a J-1 visa or C-1/D visa that he used to enter the U.S. (usually it is an H-1B or H-2B, and those are ok), then after you marry, you will need to file an adjustment package.
This includes the following forms:
I-130 (Petition for family member), I-485
(application for Lawful Permanent Residency), I-765 (Application for work permit), I-131 (Application for Advance Parole), G-325a (Biographic data - one for each of you), I-693 (Medical exam that a certified doctor must fill), and I-864 (Affidavit of support).
You will need to file each form with supporting evidence and appropriate filing fees. You can find these forms at www.uscis.gov/forms
The supporting evidence that you would file would be birth certificates, marriage certificate, divorce certificates if either of you have been married previously, proof of your U.S. Citizenship, proof of your spouse's legal entry into the U.S., and financial documents to prove your income over the last year at least.
In about 3 to 5 months after filing, he should get a work permit. About 5 to 7 months after filing you should get a marriage interview. If all goes well, a few weeks later he should get a Residency status (green card).
And just so you are clear, once those forms are properly filed with USCIS, it does not matter that his I-94 (work status) expires. He would still be in legal status authorized by the Attorney General as long as that I-485 is pending. He does not go out of status or accrue illegal time while that is pending. So his employer does not have to file anything for him if they do not want to.
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