Do I have to admit to a prior DUI?

Yes. Or else you could open yourself up to further prosecution.

If you are directly asked by the court or the prosecutor, you must answer truthfully. Otherwise, you risk being charged with obstruction of justice/another crime.

But if they do not ask, you do not have any obligation to tell.

Good luck!

I would add to the two above answers in this manner.
If ask by the Judge in Court, the answer is yes.

If ask by the prosecutor, he/she probably knows the answer already, but are setting a trap so they can show the Judge that you are dishonest if you lie about the previous DUI.

The main problem with your question is not enough facts. 1999 DUI; what state? Some states do have diversion programs that remove the arrest and conviction after successful completion.

If this is for an alcohol evaluation counselor and you know for a fact that it was diverted and dismissed, then you wouldn’t have to admit it and if they found out, what would happen? You would be at the same level that you are trying to avoid anyway.
Good luck.