IF I were to build a bike from scratch i would build one of these http://www.rqriley.com/bike.html
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Also, watch for police auctions at which unclaimed property is sold. Go to Goodwill stores, yard sales, etc. Let people know you are looking for a bike. Someone probably has one gathering dust that they would sell cheap or give you. Go to scrap yards. Often bicycle carcasses end up there. You can buy the various parts you need pretty cheap and build your own. What you get may not have the newest and lightest frame, but it will be a decent bike when you are finished tuning it up.
Take a careful look at how a 10-speed (more commonly 12 or 18 now) bike and see what you can reproduce. A welder isn't going to help much, there are a few fixed joints in the frame, but the working parts need machining.
If you have a welder, make something and sell it or barter for a bike. I frequently see 10-speed bikes for $20 at garage sales.
Buying the steel to weld into a frame, plus the components for a 10-speed bike, is likely to become quite expensive. If you're really trying to save money, I'd advise checking something like Craigslist or Freecycle for a used ten-speed and fixing it up. There are plenty of Instructables about fixing and tuning up a bike - check those out for help.