The T stays the same but the Entropy takes a huge jump. At the T of fusion, the solid and liquid components are in equilibrium. The next portion of the curve is the increase in Entropy with T for the liquid phase. This continues until we reach T vap , the Temperature at which the liquid vaporizes to the gaseous state. This is the largest change in Entropy, because of the enormous difference in the molar volumes of liquids versus gases.
S o is non-zero for elements, because elements have heat capacities.