Got to agree with Jazz that arc is much better than mig for this sort of thing.

Something you may want to consider is having a look on the skill swop websites to see if you can find someone local offering free welding or just find someone that you can pay to do the one off welding job for you. They will probably do a much better job in a couple of hours for beer money, so if you only have this one job that needs welding it might be a lot cheaper and easier to outsource it.

Buying your own welder even second hand, some sticks, gloves and a mask mean you are probably going to drop £50 - £150 on gear easily and then you will spend hours researching how to use the thing, hours practicing and then do a not great job of welding the frame.

Of course if you want to learn to weld and have lots of ideas for projects that need welding, it's lots of fun and very satisfying so go for it! :)