Supersolids are exotic materials combining the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid. The supersolid phase of matter was predicted 50 years ago(1-3) for solid helium(4-8). Ultracold quantum gases have recently been made to exhibit periodic order typical of a crystal, owing to various types of controllable interaction(9-13). A crucial feature of a D-dimensional supersolid is the occurrence of D + 1 gapless excitations, reflecting the Goldstone modes associated with the spontaneous breaking of two continuous symmetries: the breaking of phase invariance, corresponding to the locking of the phase of the atomic wave functions at the origin of superfluid phenomena, and the breaking of translational invariance due to the lattice structure of the system. Such modes have been the object of intense theoretical investigations(1,14-18), but they have not yet been observed experimentally. Here we demonstrate supersolid symmetry breaking through the appearance of two distinct compressional oscillation modes in a harmonically trapped dipolar Bose-Einstein condensate, reflecting the gapless Goldstone excitations of the homogeneous system. We observe that the higher-frequency mode is associated with an oscillation of the periodicity of the emergent lattice and the lower-frequency mode characterizes the superfluid oscillations. This work also suggests the presence of two separate quantum phase transitions between the superfluid, supersolid and solid-like configurations.