The constraints imposed by the high spin rate of PSR 1937+214 on the masses and radii of nonrotating neutron stars that can be spun up to such a short period are examined. Assuming uniform rotation, it is found that none of the currently viable nuclear equations of state can be ruled out. However, the discovery of a single pulsar with period of less than about 0.5 ms would rule out all current equations of state. Assuming that PSR 1937+214 was spun up uniformly from a nonrotating progenitor requires that the maximum mass of a nonrotating star in general relativity cannot exceed 12.5 solar masses, independent of any knowledge of the equation of state. This limit decreases linearly with the shortest known period of a pulsar.