The other posts have lots of good ideas. Here's one that takes off from what Krishna-agrawala said about putting yourself in situations where you must interact with people you don't know.
In my opinion, it's not so much speaking in public that you're afraid of so much as just interacting with strangers. And so it's not just speaking but interacting that you should practice.
I would suggest trying to get a job (or volunteer opportunity) where you would interact with the public. My wife was painfully shy through college. Then she started to work as a retail clerk where she had to interact with strangers constantly.
She is now in a job where she has to address city council meetings and such (just a very small city, but still -- it's public speaking) and she does well.
Finally, Rome wasn't built in a day. Be patient with yourself.
I think one of the way that you can build confidence is when you learn how to fail miserably and still have the ability to hold your head up high. I tell you this from experience. I do a lot of public speaking and when I first started, I was pretty bad. But what got me over this is when I learned to fail well. It was actually very liberating. When I got over the fear of failure, it took off a lot of the pressure and anxiety. This made me into a better speaker. Now the question is this, how do we do this? You have to find your identity is something else. If it is in public speaking, and you fail in it, then there is little hope. But if you find your identity in something else, and you fail in public speaking, it is not really a big deal. In short, you need to find you identity is something. I found my in Christianity.
Shyness in public speaking can be resolved with practice, repetition, and commitment. The most important element to keep in mind would be that you are in control and there has to be a mentality of seizing that particular moment and owning it as your own. Ensuring that you know exactly what you are going to say would be a part of this. Memorization of the speech is essential. The next step would be to practice in front of a full length mirror to be able to note how you look to others. Note mannerisms, gestures, facial expressions that have to be modified or eliminated. Along these lines, videotaping yourself can help you understand where your physical presence needs improvement as well as the spoken quality of the speech. The more you are able to see and hear yourself give the speech, the more you will feel comfortable delivering it in public because it will simply be a repetition of what has been done many times before. Additionally, practicing it in front of a few close friends or a small group might also help. Little, if anything, can be done to eliminate all of the fear and concerns with delivering the speech in front of a large group, but some intermediary steps can be taken to minimize its overall impact.
In trying to overcome shyness it is very important to realize that shyness is an essential and even useful quality in all people. So shyness is not something that you can, or even need to completely eliminate from your life. What you need to is to manage your shyness and overcome it in situations where it is necessary to do so.
Second thing, shyness and confidence are not opposites, although nervousness at times may be considered opposite of confidence. It is important for you to be clear in your mind about what is your problem. Nervousness is caused by a feeling of insecurity, by a fear of being unable to handle a situation appropriately. It is possible to overcome by being more thorough in your preparation and by progressively taking on more responsibilities, and performing more difficult tasks.
Shyness refers specifically to the feeling of hesitation in interacting with others or otherwise exposing your thoughts and behavior to others. In public speaking the effect of shyness gets multiplied because of the presence of a large number of people, and for the need of public speaker to take the initiative and act like a central figure in the whole proceedings.
To overcome your shyness and to act confidently in presence of others, it not to fight your shyness. When you feel shy, remind yourself that it is a natural reaction for all the people, and in this way accept your shyness, rather than fight it. If you try to fight your shyness, this can make you nervous and increase your problem. Give yourself a few seconds to let the feeling of shyness settle down. Deep breathing and taking a sip of water can also help.
Second, approach to overcoming shyness is to practice in small steps interacting with people you don't know. For situations like these it is good to have ready with you as repertoire of subjects for small talk.
Public speaking is more than just overcoming your shyness. You need to develop the ability to think on your feet, and to express your ideas in a clear and interesting way. These two abilities are best developed with practice. You nod to start practicing with making short speeches, may be just one minute speech, on subjects you are familiar with. Then you can gradually move to longer speech on a wader range of subjects.
It would be best if you an form a small group of friends to practice public speaking this way. Even a group of two can be very effective. If even that is not possible, you can try practicing by yourself.