  We investigate the consequences of resonant tunneling of Cooper pairs on the
quantum phase slips occurring in a Josephson junction. The amplitude for
quantum tunneling under the Josephson potential barrier is modified by the
Landau-Zener amplitude of adiabatic passage through an Andreev level crossing,
resulting in the suppression of $2\pi$ phase slips. As a consequence, close to
resonance, $4\pi$ phase slips become the dominant tunneling process. We
illustrate this crossover by determining the energy spectrum of a transmon
circuit, showing that a residual charge dispersion persists even at perfect
transparency.
