  We compute the quantum string entropy S_s(m, H) from the microscopic string
density of states of mass m in Anti de Sitter space-time. For high m, (high Hm
-->c/\alpha'), no phase transition occurs at the Anti de Sitter string
temperature T_{s} which is higher than the flat space (Hagedorn) temperature
t_{s}. (the Hubble constant H acts as producing a smaller string constant and
thus, a higher tension). T_s is the precise quantum dual of the semiclassical
(QFT) Anti de Sitter temperature scale . We compute the quantum string emission
by a black hole in Anti de Sitter space-time (bhAdS). In the early evaporation
stage, it shows the QFT Hawking emission with temperature T_{sem~bhAdS},
(semiclassical regime). For T_{sem~bhAdS}--> T_{s}, it exhibits a phase
transition into a Anti de Sitter string state. New string bounds on the black
hole emerge in the bhAdS string regime. We find a new formula for the full
(quantum regime included) Anti de Sitter entropy S_{sem}, as a function of the
usual Bekenstein-Hawking entropy S_{sem}^(0). For low H (semiclassical regime),
S_{sem}^(0) is the leading term but for high H (quantum regime), no phase
transition operates, in contrast to de Sitter space, and the entropy S_{sem} is
very different from the Bekenstein-Hawking term S_{sem}^(0).
