Your recollection is close, but not quite right.
In the "pasha" act, Rick (who was heavy set with a beard) wore a caftan and a fez. His assistant (played by his wife) wore some sort of dress; it was not "skimpy."
The act was structured thusly: Rick would perform (silently) to music ("The Baby Elephant Walk"). Every now and then, the music would abruptly change to something more ethereal. Rick would freeze in position, while the assistant brought out a new signboard announcing the next trick, and placed it onto an easel. She'd leave, the "Elephant" music would resume, and Rick would reanimate and proceed with the next routine.
After his wife brought on the "Balloon to Cannonball" sign, when his music resumed, Rick would look at the sign, then act as if the black balloon had suddenly gotten very heavy (i.e., turned into a cannonball). This got great laughs, as the audience assumed it was a mime gag. Then he'd pluck off the tied end of the balloon, and drop the ball to the stage, where it landed with a resouding thunk.
The reason I remember this clearly is in part because Rick was a good friend, but also because I was inadvertently the source of the trick. At the Tannen Jubilee in 1974, Paul Curry told me of several unsolved plots he'd devised over the years. One that particularly amused me was this: Magician inflates balloon. He takes a pin, pops the balloon, and out falls a bowling ball.
I thought this image was hysterical, and with Paul's permission I described it to Rick, with no thought that he'd ever attempt to solve it. Less than a year later, he had developed the "Balloon to Cannonball" variation.