Supersolidity combines superfluid flow with long-range spatial periodicity of solids(1), two properties that are often mutually exclusive. The original discussion of quantum crystals(2) and supersolidity focused on solid He-4 and triggered extensive experimental efforts(3,4) that, instead of supersolidity, revealed exotic phenomena including quantum plasticity and mass supertransport(4). The concept of supersolidity was then generalized from quantum crystals to other superfluid systems that break continuous translational symmetry. Bose-Einstein condensates with spin-orbit coupling are predicted to possess a stripe phase(5-7) with supersolid properties(8,9). Despite several recent studies of the miscibility of the spin components of such a condensate(10-12), the presence of stripes has not been detected. Here we observe the predicted density modulation of this stripe phase using Bragg reflection (which provides evidence for spontaneous long-range order in one direction) while maintaining a sharp momentum distribution (the hallmark of superfluid Bose-Einstein condensates). Our work thus establishes a system with continuous symmetry-breaking properties, associated collective excitations and superfluid behaviour.