  We compute the quantum string entropy S_s(m, H) from the microscopic string
density of states rho_s (m,H) of mass m in de Sitter space-time. We find for
high m, a {\bf new} phase transition at the critical string temperature T_s=
(1/2 pi k_B)L c^2/alpha', higher than the flat space (Hagedorn) temperature
t_s. (L = c/H, the Hubble constant H acts at the transition as producing a
smaller string constant alpha' and thus, a higher tension). T_s is the precise
quantum dual of the semiclassical (QFT Hawking-Gibbons) de Sitter temperature
T_sem = hbar c /(2\pi k_B L). We find a new formula for the full de Sitter
entropy S_sem (H), as a function of the usual Bekenstein-Hawking entropy
S_sem^(0)(H). For L << l_{Planck}, ie. for low H << c/l_Planck,
S_{sem}^{(0)}(H) is the leading term, but for high H near c/l_Planck, a new
phase transition operates and the whole entropy S_sem (H) is drastically
different from the Bekenstein-Hawking entropy S_sem^(0)(H). We compute the
string quantum emission cross section by a black hole in de Sitter (or
asymptotically de Sitter) space-time (bhdS). For T_sem ~ bhdS << T_s, (early
evaporation stage), it shows the QFT Hawking emission with temperature T_sem ~
bhdS, (semiclassical regime). For T_sem ~ bhdS near T_{s}, it exhibits a phase
transition into a string de Sitter state of size L_s = l_s^2/L}, l_s=
\sqrt{\hbar alpha'/c), and string de Sitter temperature T_s. Instead of
featuring a single pole singularity in the temperature (Carlitz transition), it
features a square root branch point (de Vega-Sanchez transition). New bounds on
the black hole radius r_g emerge in the bhdS string regime: it can become r_g =
L_s/2, or it can reach a more quantum value, r_g = 0.365 l_s.
