The electrical properties of individual bundles, or "ropes," of single-walled carbon nanotubes have been measured. Below about 10 kelvin, the low-bias conductance was suppressed for voltages less than a few millivolts. In addition, dramatic peaks were observed in the conductance as a function of a gate voltage that modulated the number of electrons in the rope. These results are interpreted in terms of single-electron charging and resonant tunneling through the quantized energy levels of the nanotubes composing the rope.