We report a high-resolution x-ray diffraction study of the freezing transitions of krypton on graphite for coverages above one monolayer. For pressure P>430 Torr there is a single continuous incommensurate-fluid-to- incommensurate-solid freezing transition. For P<430 Torr the freezing at fixed pressure is doubly reentrant with the succession of phases fluid , commensurate solid , reentrant fluid , incommensurate solid. The first transition is of first order, whereas the other two are continuous within the experimental resolution. © 1984 The American Physical Society.