How to butt weld 2 pipes together straight?

My Toad V8 is running well right now and doesn't seem to have any issues (knock on wood), so I am using it as a driver and have two of my other vehicles under restoration. Both of those cars have their wheels removed, but I need to be able to move them around the shop in order to install overhead lights, so I need to build a steel frame on casters which will attach to these cars' axles. I can do this OK, except that my garage floor is NOT flat, and I need to butt weld two pieces of pipe together to make one long, straight piece. How to make sure they are straight before welding? Clamp something straight to them, obviously, but I am not sure how I would do this in both the horizontal and vertical planes at the same time, and still have room to heat the pipe and weld it. Also, isn't all the stuff I am going to clamp to hold the pipes straight going to act as a big heat sink?

I have been butt-welding pipe together for a long time, but I've never figured out how to weld it straight. I know I really should clamp it to a machinist's table, but don't hold your breath until I get one of those. Suggestions would be welcome. A simple jig, maybe? Thanks a lot.