The identification of brain regions that are associated with the conscious perception of visual stimuli is a major goal in neuroscience. Here we present a test of whether the signals on neurons in cortical area V1 correspond directly to our conscious perception of binocular stereoscopic depth. Depth perception requires that image features on one retina are first matched with appropriate features on the other retina. The mechanisms that perform this matching can be examined by using random-dot stereograms, in which the left and right eyes view randomly positioned but binocularly correlated dots. We exploit the fact that anticorrelated random-dot stereograms (in which dots in one eye are matched geometrically to dots of the opposite contrast in the other eye) do not give rise to the perception of depth because the matching process does not find a consistent solution. Anti-correlated random-dot stereograms contain binocular features that could excite neurons that have not solved the correspondence problem. We demonstrate that disparity-selective neurons in V1 signal the disparity of anticorrelated random-dot stereograms, indicating that they do not unambiguously signal stereoscopic depth. Hence single V1 neurons cannot account for the conscious perception of stereopsis, although combining the outputs of many V1 neurons could solve the matching problem. The accompanying paper suggests an additional function for disparity signals from V1: they may be important for the rapid involuntary control of vergence eye movements (eye movements that bring the images on the two foveae into register).