immigration law, procedure for adjusting illegal alien's status

Unless you are now married to the alien, there is nothing for you to do to "fix his status".

If you two are now married with each other, the next question is whether you are a US citizen. If you are not a US citizen, your husband cannot adjust status to a lawful permanent resident based on your marriage unless he is in legal status. Moreover, if you are only a green card holder, there is a long wait before a visa number becomes available to your husband.

If you are a US citizen, whether your husband can adjust status depends on what you mean by "illegal". If he was inspected when he last entered the US and simply overstayed his authorized stay, his marriage to a US citizen would allow him to adjust status while remaining in the US provided he has no other immigration problem.

If he came into the US without being inspected by the government, he cannot adjust status while remaining in the US unless he has an approvable petition filed for him before a certain date in 2001.

Besides the inspection issue, there are many other reasons why an alien would not be granted permission to stay in the US.

Your best option may be to review the specific facts with an attorney. Alternatively, use the search function of this site and search for the words "adjust status" for the prior posts with similar issues.