We propose a novel type of composite light-matter interferometer based on a supersolid-like phase of a driven Bose-Einstein condensate coupled to a pair of degenerate counterpropagating electromagnetic modes of an optical ring cavity. The supersolid-like condensate under the influence of the gravity drags the cavity optical potential with itself, thereby changing the relative phase of the two cavity electromagnetic fields. Monitoring the phase evolution of the cavity output fields thus allows for a nondestructive measurement of the gravitational acceleration. We show that the sensitivity of the proposed gravimeter exhibits Heisenberg-like scaling with respect to the atom number. As the relative phase of the cavity fields is insensitive to photon losses, the gravimeter is robust against these deleterious effects. For state-of-the-art experimental parameters, the relative sensitivity ∆g/g of such a gravimeter could be of the order of 10 −10 -10 −8 for a condensate of a half a million atoms and interrogation time of the order of a few seconds.Introduction.-Precision measurement plays a vital role in fundamental sciences as well as technological applications. Notably, at the beginning of the twentieth century discrepancies between precise measurements and theory led to the birth of quantum mechanics [1]. Interestingly, quantum mechanics itself in turn opened an entirely new avenue in precision measurement. One of its most flourishing branches is quantum metrology, which exploits the quantum-mechanical framework to perform even more precise measurements than it is allowed by classical approaches [2,3]. Remarkable examples include the development of precise "gravimeters" based on quantum mechanical effects.A gravimeter is an apparatus that measures the local gravitational acceleration. It allows to measure, e.g., magma build-up before volcanic eruptions, hidden hydrocarbon reserves, and Earth's tides [4]. In addition, it also allows to test more fundamental aspects of physics such as local Lorentz invariance [5], the isotropy of post-Newtonian gravity [6], and quantum gravity [7]. The current generation of gravimeters include: microelectromechanical gravimeters [4], free-fall gravimeters [8][9][10][11][12], spring-based gravimeters [13,14], superconducting gravimeters [15], optomechanical gravimeters [16,17], and atom interferometers [18][19][20][21][22][23].In the above list, the atom interferometry deserves a special position because of the possibility of harnessing quantum features of many-body systems [24]. In principle, by using entangled resources, it is possible to increase the precision of measurement over the shot-noise limit [25]. However, noise and decoherence limit the creation and use of quantum correlations [26,27], especially for large samples. Hence, sub-shot-noise interferometry is currently restricted to proof-of-principle experiments with atoms [28][29][30][31][32][33][34] as well as photons [35][36][37][38][39].In this Letter, we propose a novel type of gravimeter based on a supersolid-like state of...