  We report results of follow-up multicolor photometry of the optical variable
source that is a probable remnant of the gamma-ray burst GRB970508 discovered
by the BeppoSAX satellite (IAUC 6649). Observations were carried out in
Johnson-Kron-Cousins $ BVR_cI_c system with the 1-m and 6-m telescopes of SAO
RAS. Between the 2nd and the 5th day after the burst a fading of the remnant is
well fitted with an exponential law in all four bands. During this period the
`broadband spectrum' of the object was unchanged and can be approximated by a
power-law,
  F_{\nu}\propto\nu^{-1.1}. After the 5th day the decline of brightness is
slowed down. In the R_c band until the 32nd day, the light curve can be
described by a power-law relation, F_t \propto t^{-1.2}.
