  The vortex phase diagram in the external rotation frequency versus
temperature is calculated for dilute Bose-Einstein condensed gases. It is
determined within the Bogoliubov-Popov theory for a finite temperature where
the condensate and non-condensate fractions are treated in an equal footing.
The temperature dependences of various thermodynamic instability lines for the
vortex nucleation are computed to construct the phase diagram. Experiments are
proposed to resolve a recent controversy on the vortex creation problem
associated with the quantized vortex observation in $^{87}$Rb atom gases.
