This article describes a new numerical scheme to model surface tension for an interface represented by a level-set function. In contrast with previous schemes, the method conserves fluid momentum and recovers Laplace's equilibrium exactly. It is formally consistent and does not require the introduction of an arbitrary interface thickness, as is classically done when approximating surface-to-volume operators using Dirac functions. Variable surface tension is naturally taken into account by the scheme and accurate solutions are obtained for thermocapillary flows. Application to the Marangoni breakup of an axisymmetric droplet shows that the method is robust also in the case of changes in the interface topology.