Despite all the progress of semiconductor integrated circuit technology, the extreme complexity of the human cerebral cortex makes the hardware implementation of neuromorphic networks with a comparable number of devices exceptionally challenging. One of the most prospective candidates to provide comparable complexity, while operating much faster and with manageable power dissipation, are so-called CrossNets based on hybrid CMOS/memristor circuits. In these circuits, the usual CMOS stack is augmented with one or several crossbar layers, with adjustable two-terminal memristors at each crosspoint. Recently, there was a significant progress in improvement of technology of fabrication of such memristive crossbars and their integration with CMOS circuits, including first demonstrations of their vertical integration. Separately, there have been several demonstrations of discrete memristors as artificial synapses for neuromorphic networks. Very recently such experiments were extended to crossbar arrays of phase-change memristive devices. The adjustment of such devices, however, requires an additional transistor at each crosspoint, and hence the prospects of their scaling are less impressive than those of metal-oxide memristors, whose nonlinear I-V curves enable transistor-free operation. Here we report the first experimental implementation of a transistor-free metal-oxide memristor crossbar with device variability lowered sufficiently to demonstrate a successful operation of a simple integrated neural network, a single layer-perceptron. The network could be taught in situ using a coarse-grain variety of the delta-rule algorithm to perform the perfect classification of 3x3-pixel black/white images into 3 classes. We believe that this demonstration is an important step towards the implementation of much larger and more complex memristive neuromorphic networks.