I want to be a web developer, and everyday I ask myself the same question: why am I wasting my time getting a computer science degree?
If your goal in life is to edit HTML files, then don't go get a degree at all. Like the (only 2 at the time of my writing this) other posters have said, build and run a website or 3, get a job as an intern at a company that does technology stuff, do things on your own and build a portfolio.
That being said, a career as a web designer is good for about 1 or 2 steps up, and then you are left as...a very experienced web designer, with a resume of short-term contract-type positions, and no educational background to show you know anything about databases, or actual programming, or business, or being able to make it through a degree program, or being able to deal with being around people, or anything much to give me or any other hiring manager a reason to spend more than 30 seconds reading your resume.
Just last week, I tossed a resume, for a new DBA position we're trying to fill, in the trash. Maybe he was a good DBA, but his resume was 10 years of web development and website administration, with 2 6-month stints at a place being an actual DBA in any sense of the word. Yes, I know any web developer that deals with running websites most likely has run the databases that back those websites, especially at a smaller company (I've done it myself at even not-so-small companies), but I can't suggest hiring someone to manage 2 dozen customer databases in a production environment with 12 months of short-lived "DBA" experience on a 10 year career in technology.
If you want to be a "Web Designer" for all of eternity, then teach yourself and build your portfolio of Things You've Done to show off to potential customers.
If you want a career in technology that might eventually lead to managing and hiring other web developers, or moving into Production Operations, or dealing with technology workers that do anything other than develop web pages, get a degree. Get a business or management or sociology degree if you think a CompSci degree will be a "waste", but give future hiring managers a reason to think you have the ability to learn more than how to monkey with HTML.