Using the Polchinski-Strominger effective string theory in covariant gauge, we compute the mass of a rotating string in D dimensions with large angular momenta J, in one or two planes, in fixed ratio, up to and including first subleading order in the large J expansion. This constitutes a firstprinciples calculation of the value for the order-J 0 contribution to the mass-squared of a meson on the leading Regge trajectory in planar QCD with bosonic quarks. For open strings with Neumann boundary conditions, and for closed strings in D ≥ 5, the order-J 0 term in the mass-squared is exactly calculated by the semiclassical approximation. This term in the expansion is universal and independent of the details of the theory, assuming only D-dimensional Poincaré invariance and the absence of other infinite-range excitations on the string worldvolume, beyond the Nambu-Goldstone bosons.