If you've just got one or two pieces, you should begin by playing them for a close friend. It's important that the friend makes you nervous, so be careful about whom you ask. Chances are that you won't be happy with that one performance, so think about playing the same pieces for that friend a week later. Whatever you do, try to replicate the performance experience as much as possible. Be formal, and make sure that your friend is an attentive listener. Value the criticism that the friend gives you, but don't be too hard on yourself.
When you're more confident with your piece or pieces, see if you can schedule them for a relatively low-impact event. Perhaps you could play for friends at a small party, or at a church service, or within the context of a larger chamber music recital. Be careful about some of these choices, however.
Parties may not be the best place, especially if people don't really want to listen or (gasp!) you've been drinking. If you're playing for a church service--especially with a newly memorized piece--, make sure that it's scheduled as a prelude or early in the service. All of the church service downtime (hymns, readings, sermon, prayers) will build up a whole lot of anxiety, and by the time it's your turn to play, you might be in a very bad place. Be careful as well about playing a solo piece on a chamber music program. It's sometimes very hard to change from reading-music-mind to playing-from-memory-mind. In short, think hard about where and when you're going to perform.
At times, I've used Google+ to create live streams of my practice performances. I've done this for individual works and also for full concerts. With Google+, I don't even need to invite an audience. They just show up when I create a public stream! A few years ago, one of my recitals was cancelled due to snow, so I broadcast the recital via a live stream on blogTV. A local newspaper picked this up, and I had 50 viewers at one point. You can be assured that the pressures of a live stream are about the same as for a performance in front of an audience. (I only missed having audience feedback, but I did hold up applause signs!)
And now what you've been waiting for: Yes, you can put your piece to rest! Congratulations! You've now graduated from the Middle Stages of Memory.