An unshunted Josephson tunnel junction switching out of its zero-voltage state is a model system for thermal activation out of a metastable state. For small-capacitance low-critical-current junctions, this thermal activation process follows a generalized Arrhenius law involving dissipation directly in its exponent. This escape over a dissipation barrier can be computed exactly for a junction connected to a {ital RC} impedance providing large damping. The diffusion branch and the switching histograms measured for such a junction are in agreement with theory. {copyright} {ital 1996 The American Physical Society.}