  Brightness distribution of Gamma-Ray Bursts (GRBs) is studied in detail under
the assumption that GRB rate is related to cosmic star formation rate. The two
populations of the long- and short-duration bursts in the 4B BATSE catalog are
analyzed separately. Taking account of current uncertainties in the
observational estimate of star formation rate (SFR), we have tried various
models of the cosmic star formation history and we find that the SFR evolution
in $z$ = 0--1 is strongly constrained by the GRB distribution if the standard
candle approximation is valid. The strong SFR evolution by a factor of $\sim$
15 from $z$ = 0 to 1 inferred from UV observations is too steep to be
consistent with the GRB distribution for any distance scale of GRBs. Some
possibilities to reconcile this discrepancy are discussed, including the
intrinsic luminosity dispersion of GRBs and/or modification of star formation
history estimated by UV observations. We argue that SFR increase factor from
$z$ = 0 to 1 may be as low as about 4 if we choose different sets of
cosmological parameters and/or take account of the evolution of metallicity and
dust extinction in the UV data, and this would significantly remedy the
discrepancy.
