converting a nine speed hybrid disc to a seven speed drop bar, is it possible? I have a used Jamis Coda Elite that I'd love to convert to a drop bar bike, but on the cheap.  I also have an older drop bar mountain bike that already has the 7 speed brifters on it.

Can I just swap out the rear cassette on the Jamis for the cassette on the MTB?  Will the rim on the older MTB have room for the disc swap?

The front tires would stay the same; just new cable to the brake for the brifters. Any help would be greatly appreciated.
The Coda's freehub body can take a 7-speed cassette with the addition of a 4.5mm conversion spacer to take up the extra room, which is a common bike shop item.

There are two stumbling blocks though.

First, if the Coda's current left shifter and front derailer come from a Shimano mountain group, which they probably do, then they won't work correctly with STI because the actuation ratio is different between mountain and road. It can be made to work badly but will never be right. So if this is what you have (as opposed to a flat bar road shifter and a road derailer), you'll need a road derailer, and then there's the question of getting one that either plays nice with the existing crank or getting a crank to match it. If you don't want to have to mess with changing cranks, you may look at the IRD Alpina FD, because it's just about the only one out there that has a cage profiled for mountain/hybrid/trekking/etc triples and takes road cable pull.

Second, you probably have normal v-brake-lever-compatible brake calipers, whereas your STIs have road/sidepull/canti cable pull. They don't work together, at all. So you need either road calipers or a cable pull conversion device such as Travel Agents.