It is shown that it is possible to perform combined X-ray and neutron single-crystal studies in the same diamond anvil cell (DAC). A modified Merrill-Bassett DAC equipped with an inflatable membrane filled with He gas has been developed. It can be used on laboratory X-ray and synchrotron diffractometers as well as on neutron instruments. The data processing procedures and a joint structural refinement of the high-pressure synchrotron and neutron single-crystal data are presented and discussed for the first time.