A supersolid is a counter-intuitive state of matter that combines the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid1,2. Since the first prediction3 in the 1950s, experimental efforts to realize this state have focused mainly on helium, in which supersolidity remains unobserved4. Recently, supersolidity has also been studied in ultracold quantum gases, and some of its defining properties have been induced in spin–orbit-coupled Bose–Einstein condensates (BECs)5,6 and BECs coupled to two crossed optical cavities7,8. However, no propagating phonon modes have been observed in either system. Recently, two of the three hallmark properties of a supersolid—periodic density modulation and simultaneous global phase coherence—have been observed in arrays of dipolar quantum droplets9,10,11, where the crystallization happens in a self-organized manner owing to intrinsic interactions. Here we directly observe the low-energy Goldstone mode, revealing the phase rigidity of the system and thus proving that these droplet arrays are truly supersolid. The dynamics of this mode is reminiscent of the effect of second sound in other superfluid systems12,13 and features an out-of-phase oscillation of the crystal array and the superfluid density. This mode exists only as a result of the phase rigidity of the experimentally realized state, and therefore confirms the superfluidity of the supersolid.