The full H-T phase diagram in the nematic superconductor FeSe is mapped out using specific-heat and thermal-expansion measurements down to 0.7 K and up to 30 T for both field directions. A clear thermodynamic signal of an underlying vortex-melting transition is found in both datasets and could be followed down to low temperatures. The existence of significant Gaussian thermal superconducting fluctuations is demonstrated by ascaling analysis, which also yields the mean-field upper critical field Hc2(T). For both field orientations, Hc2(T) shows Pauli-limiting behavior. Whereas the temperature dependence of the vortex-melting line is well described by the model of Houghton et al., Phys. Rev. B 40, 6763 (1989) down to the lowest temperatures for H ? FeSe layers, the vortex-melting line exhibits an unusual behavior for fields parallel to the planes, where the Pauli limitation is much stronger. Here, the vortex-melting anomaly is only observed down to T*=2-3 K, and then merges with the Hc2(T) line as predicted by Adachi and Ikeda, Phys. Rev. B 68 184510 (2003). Below T, Hc2(T) also exhibits a slight upturn possibly related to the occurence of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state.