jmc

At 3:20, the minute hand is at the 4, $\frac{1}{3}$ of the way around the clock, therefore, $120$ degrees from the top. The hour hand, at the beginning of the hour, started at the 3, but since we are $\frac{1}{3}$ of the way through the hour, the hour hand will have traveled $\frac{1}{3}$ the arc from 3 to 4. Therefore, the hour hand is another 10 degrees further clockwise, meaning the measure of the angle between the two hands is $120-100=\boxed{20\text{ degrees}}$.