We present Chandra and XMM-Newton X-ray observations that monitor the neutron star cooling of the quasi-persistent neutron star X-ray transients KS 1731-260 and MXB 1659-29 for approximately 4 yr after these sources returned to quiescence from prolonged outbursts. In both sources the outbursts were long enough to significantly heat the neutron star crust out of thermal equilibrium with the core. We analyse the X-ray spectra by fitting absorbed neutron star atmosphere models to the observations. The results of our analysis strengthen the preliminary findings of Wijnands et al. that in both sources the neutron star crust cools down very rapidly suggesting it has a high heat conductivity and that the neutron star core requires enhanced core cooling processes. Importantly, we now detect the flattening of the cooling in both sources as the crust returns to thermal equilibrium with the core. We measure the thermal equilbrium flux and temperature in both sources by fitting a curve that decays exponentially to a constant level. The cooling curves cannot be fit with just a simple exponential decay without the constant offset. We find the constant bolometric flux and effective temperature components to be (9.2 +/- 0.9) × 10-14ergcm-2 s-1 and 70.0 +/- 1.6 eV in KS 1731-260 and (1.7 +/- 0.3) × 10-14ergcm-2 s-1 and 51.6 +/- 1.4 eV in MXB 1659-29. We note that these values are dependent on the assumed distance to the sources and the column density which was tied between the observations due to the low number of photons in the latter observations. However, importantly, the shape of the cooling curves is independent of the distance assumed. In addition, we find that the crust of KS 1731-260 cools faster than that of MXB 1659-29 by a factor of ~2, likely due to different crustal properties. This is the first time that the cooling of a neutron star crust into thermal equilibrium with the core has been observed in such detail.