  Thermal emission during X-ray bursts is a powerful tool to determine neutron
star masses and radii, if the Eddington flux and the apparent radius in the
cooling tail can be measured accurately, and distances to the sources are
known. We propose here an improved method of determining the basic stellar
parameters using the data from the cooling phase of photospheric radius
expansion bursts covering a large range of luminosities. Because at that phase
the blackbody apparent radius depends only on the spectral hardening factor
(color-correction), we suggest to fit the theoretical dependences of the
color-correction versus flux in Eddington units to the observed variations of
the inverse square root of the apparent blackbody radius with the flux. We show
that spectral variations observed during a long photospheric radius expansion
burst from 4U 1724-307 are entirely consistent with the theoretical
expectations for the passively cooling neutron star atmospheres. Our method
allows us to determine both the Eddington flux (which is found to be smaller
than the touchdown flux by 15%) and the ratio of the stellar apparent radius to
the distance much more reliably. We then find a lower limit on the neutron star
radius of 14 km for masses below 2.2M_sun, independently of the chemical
composition. These results suggest that the matter inside neutron stars is
characterized by a stiff equation of state. We finally show that the apparent
blackbody emitting area in the cooling tails of the short bursts from 4U
1724-307 is two times smaller than that for the long burst and their evolution
does not follow the theory. This makes their usage for determination of the
neutron star parameters questionable and casts serious doubts on the results of
previous works that used for the analysis similar bursts from other sources.
[abridged]
