Can you get to the top of this hole from inside the car? This would make it easier to patch repair, as you are welding on the flat, not upside-down.
Is your hole well away from the petrol tank and fuel lines, as Scott says? It is possible to weld near tanks and lines for an experienced operator, with proper shielding, but the idea of a beginner poking about near a fuel tank with a live rod could be very dangerous, to put it mildly.
I know a mechanic who was seriously burned when welding under a car. I think it was thinners which caught fire.
I also know a qualified welder who'd done a lot of welding on cars in his spare time but managed to set fire to his own car with sparks from an oxy-acetylene torch which hit a flexible fuel line. They couldn't put the flames out, but they weren't big flames at first. They towed it out of the property and it was lucky they could do that. A little garage fire extinguisher made no impression. The car was completely burnt out. However, it could have ended much worse. I think it was a case of familiarity breeding contempt.
Obviously, people do welding on cars with petrol in the tank without problems, but the safety aspects are well worth thinking about.
I wonder if the metal isn't likely to be a bit thin for arc welding anyway, especially if you don't have a lot of experience welding thin metal with arc.
my thesis is if its rusted through on a car its too thin to stick weld. Stick welding thin stuff is manky at the best of times let alone grubby metal upside down in clise promixity to a fuel tank!
MIG is better as sometimes you'll have to put up with a bit of contamination when you're welding a rusty car, which will cause problems with the tungsten electrode on a TIG.
Working in a garage I see the lads regularly weld up old cars, and sometimes I help by keeping watch for fire in the vicinity of whats to be welded on the car.
I have welded up a car with an arc welder...not easy! Took ages longer than it should!!
Some tips:
Have a pump bottle full of water, you never know where fire will break out, so being able to squirt it into a crevice is very helpful if needed.
Pull back carpets and underlay and even remove seats if necessary, these will smoulder unnoticed otherwise.
Have someone (ideally) to watch the inside of the car if you are welding from underneath incase a 'flare up' occurs.
Grind back to good shiny metal and cut away rotten pinholed metal at least an inch into sound metal, MIG needs extremely clean metal to weld onto, where's arc CAN burn through some crap, but still clean it up well.
If you must use arc, lowest amperage and thinnest rods.
A good earth point such as the lock latch on the body will ensure good smooth welding.
Its good to put a good few days worth of practice in on similar thickness steel before letting loose on the car. Is this for an MOT or restoration?