Yes sounds exactly like the alternator. As Litespeed said, check the voltage.
But first I'd check your battery terminals, make sure they're clean and tight and corrosion free. Also check the ground cables, you can easily get to two of them. Then pull back the rubber boot on the alternator power terminal and check that it's nut is on tight **** CAUTION: disconnect the negative battery terminal before you put a tool on the alternator terminal, that boot is there for a reason. Also I'd unplug the multi-pin connector to the alternator, and plug it back in just to re-seat it.
Then check the voltage at the battery terminals.....
With car off it should be at least 12.0V, although it could be less if your battery is drained.
Start the engine and check again, you're looking for a noticeable increase in voltage, ideally about 14V.
If the alternator is dead you'll probably actually see a voltage drop, to maybe 11 or 10V with the engine on since the battery is powering the ECM and ignition instead of the alternator.
You can probably get away with driving short distances by fully charging the battery overnight, as the battery can run the ignition system for a while as you've seen. Turn off all lights, radio, A/C / heater to save power. You could probably drive a few miles to work, shut the car off and still be able to start it and drive home again. Probably. Not at night, because the lights will kill the battery. This is not ideal, as it's hard on the battery to deep cycle it.
If you need a little time, you can try tapping on the alternator metal frame a few times with a hammer, that might jar it back to life for a few miles. Don't do that unless you already know the alternator is bad, per the voltage test.