Okay, to actually _crank_ the engine for starting requires a massive current draw (i.e. hundreds of amps) from the battery, but to actually keep the car running and driving requires only a few amps. If your alternator is loafing (or has even quit altogether) you can still drive non-stop for hours before the battery finally can't scrape enough amps together to keep running.

If the battery'* in good health otherwise, it'll suck up power from a jump-start in a matter of minutes, enough to restart the car once or twice, but if the alternator is not doing its thing, you'll have another dead battery real soon.

So have the alternator bench-tested and I think you'll find it'* died. After you swap the new one in, put your battery on a charger to get its juice back _before_ starting, so you don't have to jolt your new alternator with a ton of work required to bring a half-dead battery back right away.