Fast upflows observed in the late gradual phase of an M6.8 two-ribbon flare by the Solar and Heliospheric Observatory/Coronal Diagnostic Spectrometer have provided evidence for the presence of chromospheric evaporation more than an hour after the impulsive phase of the flare. The chromospheric heating necessary to generate these upflows requires the continued injection and deposition of energy, which we presume to be provided by magnetic reconnection in the flaring corona. We investigate the nature of the transport of this energy from the reconnection site to the chromosphere by comparing the observed upflow velocities with those expected from different chromospheric heating models. A nonthermal beam of energetic electrons (>~15 keV) that is capable of generating the observed velocities would also generate significant hard X-ray emission that is not observed at this stage of the flare. We conclude, therefore, that the most likely energy transport mechanism is thermal conduction.