Earth-based telescopic observations indicate that Saturn's rings are about 1 kilometer thick, while spacecraft measurements and theoretical considerations give an upper bound of about 100 meters. Analysis of a shielding effect present in radio occultation provides a sensitive new measure of the ring thickness. On the basis of this effect, Voyager 1 microwave measurements of near-forward scatter imply a thickness ranging from less than 10 meters in ring C to about 20 and 50 meters in the Cassini division and ring A, respectively. Monolayer models do not fit the observations in the latter two regions. The discrepancy between the Earth-based and spacecraft measurements may be due to warps in the ring plane or effects of tenuous material outside the primary ring system.