  In recent years increasing evidence has emerged for a thermal component in
the gamma- and X-ray spectrum of the prompt emission phase in gamma-ray bursts.
The temperature and flux of the thermal component show a characteristic break
in the temporal behavior after a few seconds. We show here, that measurements
of the temperature and flux of the thermal component at early times (before the
break) allow the determination of the values of two of the least restricted
fireball model parameters: the size at the base of the flow and the outflow
bulk Lorentz factor. Relying on the thermal emission component only, this
measurement is insensitive to the inherent uncertainties of previous estimates
of the bulk motion Lorentz factor. We give specific examples of the use of this
method: for GRB970828 at redshift z=0.9578, we show that the physical size at
the base of the flow is r_0 = (2.9+-1.8)*10^8 Y_0^{-3/2} cm and the Lorentz
factor of the flow is Gamma = (305\+-28) Y_0^{1/4}, and for GRB990510 at
z=1.619, r_0=(1.7+-1.7)*10^8 Y_0^{-3/2} cm and Gamma=(384+-71) Y_0^{1/4}, where
Y = 1 Y_0 is the ratio between the total fireball energy and the energy emitted
in gamma- rays.
