We show that the neutron star in the transient system MXB 1659-29 has a core neutrino luminosity that substantially exceeds that of the modified Urca reactions (i.e., n +n →n +p +e-+ν¯ e and inverse) and is consistent with the direct Urca (n →p +e-+ν¯e and inverse) reaction occurring in a small fraction of the core. Observations of the thermal relaxation of the neutron star crust following 2.5 yr of accretion allow us to measure the energy deposited into the core during accretion, which is then reradiated as neutrinos, and infer the core temperature. For a nucleonic core, this requires that the nucleons are unpaired and that the proton fraction exceeds a critical value to allow the direct Urca reaction to proceed. The neutron star in MXB 1659-29 is the first with a firmly detected thermal component in its x-ray spectrum that needs a fast neutrino-cooling process. Measurements of the temperature variation of the neutron star core during quiescence would place an upper limit on the core specific heat and serve as a check on the fraction of the neutron star core in which nucleons are unpaired.