Historically, the completeness of quantum theory has been questioned using the concept of bipartite continuous-variable entanglement(1). The non-classical correlations (entanglement) between the two subsystems imply that the observables of one subsystem are determined by the measurement choice on the other, regardless of the distance between the subsystems. Nowadays, continuous-variable entanglement is regarded as an essential resource, allowing for quantum enhanced measurement resolution(2), the realization of quantum teleportation(3-5) and quantum memories(3,6), or the demonstration of the Einstein-Podolsky-Rosen paradox(1,7-9). These applications rely on techniques to manipulate and detect coherences of quantum fields, the quadratures. Whereas in optics coherent homodyne detection(10) of quadratures is a standard technique, for massive particles a corresponding method was missing. Here we report the realization of an atomic analogue to homodyne detection for the measurement of matter-wave quadratures. The application of this technique to a quantum state produced by spin-changing collisions in a Bose-Einstein condensate(11,12) reveals continuous-variable entanglement, as well as the twin-atom character of the state(13). Our results provide a rare example of continuous-variable entanglement of massive particles(6,14). The direct detection of atomic quadratures has applications not only in experimental quantum atom optics, but also for the measurement of fields in many-body systems of massive particles(15).