Ultra-thin planar heterostructures of graphene and other two-dimensional crystals have recently attracted much interest. Very high carrier mobility in a graphene-on-boron nitride assembly is now well-established, but it has been anticipated that appropriately designed hybrids could perform other tasks as well. A heterostructure of graphene and molybdenum disulphide (MoS$_2$) is expected to be sensitive to photo illumination due to the optical bandgap in MoS$_2$. Despite significant advances in device architectures with both graphene and MoS$_2$, binary graphene-MoS$_2$ hybrids have not been realized so far, and the promising opto-electronic properties of such structures remain elusive. Here we demonstrate experimentally that graphene-on-MoS$_2$ binary heterostructures display an unexpected and remarkable persistent photoconductivity under illumination of white light. The photoconductivity can not only be tuned independently with both light intensity and back gate voltage, but in response to a suitable combination of light and gate voltage pulses the device functions as a re-writable optoelectronic switch or memory. The persistent, or `ON, state shows virtually no relaxation or decay within the the experimental time scales for low and moderate photoexcitation intensity, indicating a near-perfect charge retention. A microscopic model associates the persistence with strong localization of carriers in MoS$_2$. These effects are also observable at room temperature, and with chemical vapour deposited graphene, and hence are naturally scalable for large area applications.
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