  The evolution of the comoving cosmic merger-rate density of neutron star
binaries n_c(z) is calculated using a distribution of their merging times
provided by population-synthesis computations of binary stars. We adopt an
exponential law for the star formation rate with various timescales for
different morphological types of galaxies. For elliptical galaxies also an
initial burst of star formation, lasting one Gyr, is considered. The resulting
n_c(z) of most models agree with the form n_c(z) \propto (1+z)^{1.5-2} for z <
2, which has been proposed for the source population of gamma-ray bursts.
Assuming a standard candle luminosity, the computed brightness distribution is
consistent with the BATSE results if bursts at the peak flux threshold, P = 0.4
photons cm^{-2}/s, are located at a limiting redshift of 1.9 to 3.3.
Progenitors of the systems producing gamma-ray bursts at small redshift
(bright) are likely to host in spiral galaxies and star forming regions whereas
these at high redshift (dim) reside mainly in elliptical galaxies. The location
of a burst may be up to a Mpc away from the host galaxy.
