I'm not a hiring guy, but TonyC's advice is the way to go here. I'm not sure if the event in your past is a showstopper or not. But if the company doesn't feel it can trust you today, I imagine that would be a showstopper.

I know it is an embarrassing situation for you to have to address. No one is perfect and I don't think you are particularly "soiled"....don't be embarrassed....it was what it was.. almost 2 decades ago....and you're a stand-up guy for addressing it.

Even though it is a time-worn phrase, "Honesty is the best policy", exists for a reason. By disclosing the information now, you will save yourself years of anxiety wondering if you would get found out. It is a great job....but no job is worth a loss of peace of mind in my opinion.