The Josephson effect(1) is perhaps the prototypical manifestation of macroscopic phase coherence, and forms the basis of a widely used electronic interferometer-the superconducting quantum interference device(2) (SQUID). In 1965, Maki and Griffin predicted(3) that the thermal current through a temperature-biased Josephson tunnel junction coupling two superconductors should be a stationary periodic function of the quantum phase difference between the superconductors: a temperature-biased SQUID should therefore allow heat currents to interfere(4,5), resulting in a thermal version of the electric Josephson interferometer. This phase-dependent mechanism of thermal transport has been the subject of much discussion(4,6-8) but, surprisingly, has yet to be realized experimentally. Here we investigate heat exchange between two normal metal electrodes kept at different temperatures and tunnel-coupled to each other through a thermal 'modulator' (ref. 5) in the form of a direct-current SQUID. We find that heat transport in the system is phase dependent, in agreement with the original prediction. Our Josephson heat interferometer yields magnetic-flux-dependent temperature oscillations of up to 21 millikelvin in amplitude, and provides a flux-to-temperature transfer coefficient exceeding 60 millikelvin per flux quantum at 235 millikelvin. In addition to confirming the existence of a phase-dependent thermal current unique to Josephson junctions, our results point the way towards the phase-coherent manipulation of heat in solid-state nanocircuits.