THE interstellar medium (ISM) comprises a number of components at very different densities and temperatures, from dense molecular gas at 10 K to very diffuse plasma at 10(6) K. It has generally been assumed that the warm (10(4) K) and hot (similar to 10(6) K) components are in thermal pressure equilibrium(1-3), although this has not been universally accepted(4). Here we use the discovery of a shadow in the diffuse extreme ultraviolet background radiation(5), cast by a nearby cloud of cool hydrogen, to calculate directly the thermal pressure in the nearby (less than or equal to 40 pc) hot ISM. We find that it is 20 times greater than in the warm cloud surrounding the Sun. This result directly contradicts the basic assumption of thermal pressure balance in the ISM, indicating that some unidentified force (such as confinement by magnetic fields(6)) must play an important role in the overall pressure balance.