The function of a neural circuit is determined by the details of its synaptic connections. At present, the only available method for determining a neural wiring diagram with single synapse precision--a "connectome"--is based on imaging methods that are slow, labor-intensive and expensive. Here we present SYNseq, a method for converting the connectome into a form that can exploit the speed and low cost of modern high-throughput DNA sequencing. In SYNseq, each neuron is labeled with a unique random nucleotide sequence--an RNA "barcode"--which is targeted to the synapse using engineered proteins. Barcodes in pre- and postsynaptic neurons are then associated through protein-protein crosslinking across the synapse, extracted from the tissue, and then joined into a form suitable for sequencing. Although at present the inefficiency in our hands of barcode joining precludes the widespread application of this approach, we expect that with further development SYNseq will enable tracing of complex circuits at high speed and low cost.