We have investigated a small-area Josephson tunnel junction in a superconducting quantum interference device (SQUID) geometry, biased through high-resistance thin-film leads. This configuration provides the necessary conditions for the observation of charging effects. The Josephson coupling energy (Formula presented) was varied by an external magnetic flux introduced into a SQUID loop. The current-voltage characteristics of the system were measured at low temperatures T∼80 mK and different values of (Formula presented). At a certain value of the current the Zener tunneling rate exceeds that of the dissipative relaxation and the system “jumps” from the lowest to higher Brillouin zones. This corresponds to a crossover on the I-V curve. The crossover current (Formula presented) was measured as a function of (Formula presented). Good agreement between our data and a theory of Zener tunneling with dissipation has been found.