Almost 80 years ago it was predicted that, under sufficient compression, the H-H bond in molecular hydrogen (H2) would break, forming a new, atomic, metallic, solid state of hydrogen. Reaching this predicted state experimentally has been one of the principal goals in high-pressure research for the past 30 years. Here, using in situ high-pressure Raman spectroscopy, we present evidence that at pressures greater than 325 gigapascals at 300 kelvin, H2 and hydrogen deuteride (HD) transform to a new phase—phase V. This new phase of hydrogen is characterized by substantial weakening of the vibrational Raman activity, a change in pressure dependence of the fundamental vibrational frequency and partial loss of the low-frequency excitations. We map out the domain in pressure-temperature space of the suggested phase V in H2 and HD up to 388 gigapascals at 300 kelvin, and up to 465 kelvin at 350 gigapascals; we do not observe phase V in deuterium (D2). However, we show that the transformation to phase IV′ in D2 occurs above 310 gigapascals and 300 kelvin. These values represent the largest known isotropic shift in pressure, and hence the largest possible pressure difference between the H2 and D2 phases, which implies that the appearance of phase V of D2 must occur at a pressure of above 380 gigapascals. These experimental data provide a glimpse of the physical properties of dense hydrogen above 325 gigapascals and constrain the pressure and temperature conditions at which the new phase exists. We speculate that phase V may be the precursor to the non-molecular (atomic and metallic) state of hydrogen that was predicted 80 years ago.