We study a bilayer consisting of an ordinary superconductor and a magnet with a spiral magnetic structure of the Ho type. We use a self-consistent solution of the Bogolioubov-de Gennes equations to evaluate the pair amplitude, the transition temperature, and the thermodynamic functions, namely, the free energy and entropy. We find that for a range of thicknesses of the magnetic layer the superconductivity is reentrant with temperature T: as one lowers T the system turns superconducting, and when T is further lowered it turns normal again. This behavior is reflected in the condensation free energy and the pair potential, which vanish both above the upper transition and below the lower one. The transition is strictly reentrant: the low and high temperature phases are the same. The entropy further reveals a range of temperatures where the superconducting state is less ordered than the normal one.