The co-location of memory and processing is a core principle of neuromorphic computing. A local memory device for synaptic weight storage has long been recognized as an enabling element for large-scale, high-performance neuromorphic hardware. In this work, we demonstrate programmable superconducting synapses with integrated memories for use in superconducting optoelectronic neural systems. Superconducting nanowire single-photon detectors and Josephson junctions are combined into programmable synaptic circuits that exhibit single-photon sensitivity, memory cells with more than 400 internal states, leaky integration of input spike events, and 0.4 fJ programming energies (including cooling power). These results are attractive for implementing a variety of supervised and unsupervised learning algorithms and lay the foundation for a new hardware platform optimized for large-scale spiking network accelerators.