We report the results of 19 years of Arecibo timing for two pulsars in the globular cluster NGC 5904 (M5), PSR B1516+02A (M5A) and PSR B1516+02B (M5B). This has resulted in the measurement of the proper motions of these pulsars and, by extension, that of the cluster itself. M5B is a 7.95 ms pulsar in a binary system with a >0.13 M companion and an orbital period of 6.86 days. In deep HST images, no optical counterpart is detected within ~2.5 σ of the position of the pulsar, implying that the companion is either a white dwarf or a low-mass main-sequence star. The eccentricity of the orbit (e = 0.14) has allowed a measurement of the rate of advance of periastron: = 0.0142° ± 0.0007° yr ⁻¹. We argue that it is very likely that this periastron advance is due to the effects of general relativity, the total mass of the binary system then being 2.29 ± 0.17 M. The small measured mass function implies, in a statistical sense, that a very large fraction of this total mass is contained in the pulsar: M_p = 2.08 ± 0.19 M (1 σ); there is a 5% probability that the mass of this object is <1.72 M and a 0.77% probability that 1.2 M ≤ M_p ≤ 1.44 M. Confirmation of the median mass for this neutron star would exclude most "soft" equations of state for dense neutron matter. Millisecond pulsars (MSPs) appear to have a much wider mass distribution than is found in double neutron star systems; about half of these objects are significantly more massive than 1.44 M. A possible cause is the much longer episode of mass accretion necessary to recycle a MSP, which in some cases corresponds to a much larger mass transfer.