The Oberbeck–Boussinesq approximation for unsteady motion of a drop in another fluid is considered. On the unknown interface between the liquids, the surface tension is taken into account. This problem is studied in Hölder classes of functions, where the local existence theorem for the problem is proved. The proof is based on the fact that the solvability of the problem with a temperature independent right-hand side was obtained earlier. For a given velocity vector field of the fluids, a diffraction problem is obtained for the heat equation the solvability of which is established by well-known methods. The existence of a solution to the complete problem is proved by successive approximations. Bibliography: 10 titles.