A study of the dc Josephson current between two superconducting leads in the presence of a precessingclassical spin is presented. The precession gives rise to a time-dependent tunnel potential which notonly implies different tunneling probabilities for spin-up and spin-down quasiparticles, but introducesalso a time-dependent spin-flip term. We provide an exact general analytic solution for the out-of-equilibrium steady-state permanent current between two spin-singlet superconductors as a functionof the superconducting phase difference, the precession frequency and for arbitrary junction transparency.As an application we focus on the effects of the spin-flip term alone and show that the magnitude andnature of the Josephson current are indeed strongly affected by the precession of the classical spin.