A unique surface force type apparatus that allows the investigation of a confined thin film using neutron reflection is described. The central feature of the setup consists of a solid substrate (silicon) and a flexible polymer membrane (Melinex®). We show that inflation of the membrane against the solid surface provides close and even contact between the interfaces over a large surface area. Both heavy water and air can be completely squeezed out from between the flexible film and the solid substrate, leaving them in molecular contact. The strength of confinement is controlled by the pressure used to inflate the membrane. Dust provides a small problem for this approach as it can get trapped between membrane and substrate to prevent a small part of the membrane from making good contact with the substrate. This results in the measured neutron reflectivity containing a small component of an unwanted reflection, between 10% and 20% at low confining pressures (1 bar) and between 1% and 5% at high confining pressures (5 bar). However, we show that this extra signal does not prevent good and clear information on the structure of thin films being extracted from the neutron reflectivity. The effects of confinement are illustrated with data from a poly(vinyl pyrollidone) gel layer in water, a polyelectrolyte multilayer in water, and with data from a stack of supported lipid-bilayers swollen with D2O vapor. The data demonstrates the potential of this apparatus to provide information on the structure of thin films under confinement for a known confining pressure.