We have realized a quantum phase gate operating on quantum bits carried by a single Rydberg atom and a zero- or one-photon field in a high- $Q$ cavity. The gate operation is based on the dephasing of the atom-field state produced by a full cycle of quantum Rabi oscillation. The dephasing angle, conditioned to the initial atom-field state, can be adjusted over a wide range by tuning the atom-cavity frequency difference. We demonstrate that the gate preserves qubit coherence and generates entanglement. This gate is an essential tool for the nondestructive measurement of single photons and for the manipulation of many-qubit entanglement in cavity QED.