Which gives it enough juice to start but soon as I unhook the Jumper, it dies again.
I leave it on and the car runs for a short bit then dies.
Does this mean the battery is shot?
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If it’s running while the jump box is connected by dies when disconnected, it sounds like the alternator is bad.
Maybe the battery, alternator, regulator, bad ground. Get it started, drive it to a chain auto parts store, and ask them to test it for you. If it is bad and you purchase a battery from them, they will install it for free.
This is in the US, not sure it applies elsewhere.
Possibly the battery. You should have it tested to be sure.