I found the answer to my own question. Yes, installing the Disk II card would map 256 bytes of ROM on the card into the computer's address space at $C600. At that point, there were a few different options for executing that ROM code and starting the disk booting process. From http://en.wikipedia.org/wiki/Apple_DOS#Boot_loader:
Awesome stuff! Great foresight in making all peripheral cards have 256 bytes of memory directly mapped into the Apple II's address space, and in including a machine language monitor program built into ROM to make use of it. Wozniak is a really smart guy! I should invite him over for some electronics hacking…
Later versions of the Apple II ROM contained startup code to automatically search for a Disk II card, so manually starting the disk boot was no longer necessary.
Awesome stuff! Great foresight in making all peripheral cards have 256 bytes of memory directly mapped into the Apple II's address space, and in including a machine language monitor program built into ROM to make use of it. Wozniak is a really smart guy! I should invite him over for some electronics hacking…