The enzyme UDP-Galactopyranose Mutase (UGM) catalyses the conversion of galactopyranose into galactofuranose. It is known to be critical for the survival and proliferation of several pathogenic agents, both prokaryotic and eukaryotic. Among them is Trypanosoma cruzi, the parasite responsible for Chagas' disease. Since the enzyme is not present in mammals, it appears as a promising target for the design of drugs to treat this illness. A precise knowledge of the mechanism of the catalysed reaction would be crucial to assist in such design. In this article we present a detailed study of all the putative steps of the mechanism. The study is based on QM/MM free energy calculations along properly selected reaction coordinates, and on the analysis of the main structural changes and interactions taking place at every step. The results are discussed in connection with the experimental evidence and previous theoretical studies.