The classical problem of constrained melting heat transfer of a phase change material (PCM) inside a spherical capsule was revisited experimentally in the presence of nanoscale thermal conductivity fillers. The model nano-enhanced PCM (NePCM) samples were prepared by dispersing self-synthesized graphite nanosheets (GNSs) into 1-dodecanol at various loadings up to 1% by mass. The melting experiments were carried out using an indirect method by measuring the instantaneous volume expansion upon melting. The data analysis was performed based on the homogeneous, single-component assumption for NePCM with modified thermophysical properties. It was shown that the introduction of nanofillers increases the effective thermal conductivity of NePCM, in accompaniment with an undesirable rise in viscosity. The dramatic viscosity growth, up to over 100-fold at the highest loading, deteriorates significantly the intensity of natural convection, which was identified as the dominant mode of heat transfer during constrained melting. The loss in natural convection was found to overweigh the decent enhancement in heat conduction, thus resulting in decelerated melting in the presence of nanofillers. Except for the case with the lowest heating boundary temperature, a monotonous slowing trend of melting was observed with increasing the loading.