sorry duplicate post: http://www.freerepublic.com/focus/f-news/1491158/posts
No, not to the moon. Think Geostationary orbit. Although it's not quite that simple. Like communications satellite the upper anchor of the elevator would remain fixed over a given point on the equator.
As the article indicates, this notion has been around a long time, and is well studied. The only real problem is getting a material strong enough.