Defective ignition switch?

the alternator is good.

how do you know?

did you check to see if wires are good on the battery and on alternator? or is it loose on the battery terminal(s) or forgot to take off plastic caps on the battery?

If the engine starts and runs OK every time it's jump started and won't restart on it's own then your new battery is not accepting the alternators charge. Once the engine is running the the alternator supplies power for everything. At that point you could disconnect the battery and the engine would still run fine but could not restart. That makes your new battery or it's terminal connections the prime suspect.

If the alternator were failing to the degree that it couldn't recharge the battery after one normal start it would soon become obvious that something is wrong. The battery would soon be depleted and the failing alternator wouldn't be able to keep up with the demand. Lights would be dim, the engine would begin to run rough and maybe eventually stall. It could be restarted with another jump start and run fine because the donor car's alternator is providing support but when the jumper cables are disconnected the voltage collapse due to load would start all over again. It is very unlikely the engine would run and drive for the hour and 20 min. stated above without having problems.

Check the battery cable connections, to make sure they are tight and free of corrosion. No plastic caps left on like @killerzdead said. If those things check out OK I would take the battery back to where ever it came from because it is likely defective. It should have come with a warranty. Have them load test it, see if they can get it to charge.

A big spark when connecting jumper cables from another running car indicates to me your battery voltage is low, but it doesn't really prove anything. I hope this helps.

Disconnecting the battery with the engine running could possibly do serious damage to the vehicle's electrical system.