If you want somebody to make your website look good, then you are looking for an artist. If you want somebody to make it work well, give it an appropriate aesthetic for the subject matter, and add more polish in general then you want a designer.

The first can be found on many sites including 99designs.com. Finding the latter is much harder. Lots and lots of networking is probably your best bet. As a ui designer I can say that there are no specific designer events you will find me at. Most design events focus on print design, and multimedia design is a synonym for Flash in the design world.

If you do do a job posting don't call the position web designer. Call it interface designer or ux designer. The applicants you get should be much more qualified.