The measured inward motion of Phobos provides a constraint on the tidal dissipation factor, Q, within Mars. We model viscoelastic dissipation inside a convective Mars using a modified Burgers model based on laboratory experiments on anhydrous, melt-free olivine. The model tidal Q is highly sensitive to the mantle potential temperature and grain size assumed but relatively insensitive to the bulk density and rigidity structure. Q thus provides a tight constraint on the Martian interior temperature. By fitting the observed tidal Q and tidal Love number (k2) values and requiring present-day melt generation, we estimate that for a grain size of 1 cm the current mantle potential temperature is 1625±75 K, similar to that of the Earth. This estimate is consistent with recent petrologically derived determinations of mantle potential temperature but lower than estimates in some thermal evolution models. The presence of water in the Martian mantle would reduce our estimated temperature. Our preferred mantle grain size of ≈1 cm is somewhat larger than that of the Earth's upper mantle. The predicted mantle seismic Q is about 130 and is almost independent of depth. The Martian lithosphere represents a high seismic velocity lid, which should be readily detectable with future seismological observations.