We study the interplay of controlled quantum-mechanical charge and phase fluctuations by measuring their influence on the supercurrent through a small superconducting grain. The grain is connected to leads by two small Josephson tunnel junctions. The phase fluctuations are controlled by the magnetic flux threading a SQUID that connects the grain to a large superconductor. The charge fluctuations are modulated by a voltage on a gate capacitor. As a function of gate voltage we observe a broadening of the Coulomb resonance of the supercurrent for decreasing phase fluctuations, in agreement with theoretical predictions.