ac noise in disordered conductors causes both dephasing of the electron wave functions and a dc current around a mesoscopic ring. We demonstrate that the dephasing rate τ-1ϕ in long wires and the dc current , induced by the same noise and averaged over an ensemble of small rings, are connected in a remarkably simple way: τϕ = Cβe. Here e is an electron charge, and the constant Cβ~1 depends on the Dyson symmetry class. The relationship seems to agree reasonably with experiments. This suggests that the two puzzles, anomalously large persistent current and the low-temperature saturation of the dephasing, may have a common solution.