Molecular hydrogen does not adsorb on Be(0001) surfaces, suggesting activated adsorption, as was found with other s-metal/H adsorption systems. The present study investigates the interaction of thermal H atoms with clean and oxygen-covered Be(0001) surfaces using thermal desorption spectroscopy, TDS. Atomic hydrogen adsorbs at Be and oxygen-covered Be, at clean Be surfaces with a sticking coefficient in the order of unity. Molecular H-2 desorption is observed to follow a second-order rate at least in a restricted low coverage regime. Desorption is observed around 320 (O covered Be) and 450 K (clean Be) with activation energies of 14 and 20 (+/- 2) kcal/mol, respectively. Desorption frequency factors are obtained in the range 10(-4) cm2/s. At higher coverages the apparent desorption kinetics is different at Be and O-covered Be surfaces. The adsorption is endothermic with respect to H-2/Be and above 300 K hydrogen absorption in Be occurs competitively to desorption. Absorbed H in the Be bulk is outgassed with a complicated kinetics, however, H diffusion in Be seems not to limit H outgassing from Be. The formation of a Be-hydride is not observed.