  We study theoretically the large variations of the supercurrent through a
normal dot that are induced by a small quasiparticle injection current from
normal leads connected to the dot. We find that the supercurrent decomposes
into a subgap contribution, which depends on the voltages applied to the normal
leads, as well as a contribution with opposite sign from energies outside the
gap, which is insensitive to the voltages. As the voltages gradually suppress
the subgap contribution, a critical voltage exists above which the contribution
from energies outside the gap dominates, leading to a sign reversal of the
current-phase relation, namely a transition to a so-called $\pi$-junction
behavior. We determine the critical voltage and analyze the robustness of the
effect with respect to temperature and inelastic relaxation in the dot.
