Don't be fooled into thinking that "spinning" = "effortless". If your pace is slowing when you're spinning, then go up a gear and put a little bit more oomph into it. Spinning a high cadence is really about putting an even, smooth effort into the entire pedal stroke and not just putting a heavy force behind the 1 o'clock to 6 o'clock position of your pedal stroke and slacking throughout the rest.
The best description I've heard is to imagine you're trying to scrape mud off the bottom of your shoe. That way you're concentrating on that dead-center-bottom part of the pedal stroke and consciously pulling your foot back and through what is typically a dead spot.

I agree with Sixty Fiver about the fixed gear being a great teaching tool for smoothing out your spin. I built one up over the winter and recently did my first century on it. Maxed out my cadence at 155rpm without bouncing in the saddle, where my previous top out had been 130rpm.

Another good spin drill:
- Find a flat, empty 200m straightaway
- Shift into your lowest (easiest) gear
- Staying seated, sprint that low gear as fast as you can without bouncing.
The low resistance of sprinting low gears on flats means that you really have to concentrate on the bottom and upswing phases of the pedal stroke to keep from bottoming out and bouncing in the saddle.