Herpes simplex virus type 1 (HSV-1) packages its micrometers-long double-stranded DNA genome into a nanometer-scale protein shell, termed the capsid. Upon confinement within the capsid, neighboring DNA strands experience repulsive electrostatic and hydration forces as well as bending stress associated with the tight curvature required of packaged DNA. By osmotically suppressing DNA release from HSV-1 capsids, we provide the first experimental evidence of a high internal pressure of tens of atmospheres within a eukaryotic human virus, resulting from the confined genome. Furthermore, the ejection is progressively suppressed by increasing external osmotic pressures, which reveals that internal pressure is capable of powering ejection of the entire genome from the viral capsid. Despite billions of years of evolution separating eukaryotic viruses and bacteriophages, pressure-driven DNA ejection has been conserved. This suggests it is a key mechanism for viral infection and thus presents a new target for antiviral therapies.