Almost 80 years ago it was predicted that, under sufficient compression, the H-H bond in molecular hydrogen (H-2) would break, forming a new, atomic, metallic, solid state of hydrogen(1). 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, H-2 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 H-2 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 (D-2). However, we show that the transformation to phase IV' in D-2 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 H-2 and D-2 phases, which implies that the appearance of phase V of D-2 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.