  We consider the effect of accretion of radiation in the early universe on
primordial black holes in Brans-Dicke theory. The rate of growth of a
primordial black hole due to accretion of radiation in Brans-Dicke theory is
considerably smaller than the rate of growth of the cosmological horizon, thus
making available sufficient radiation density for the black hole to accrete
causally. We show that accretion of radiation by Brans-Dicke black holes
overrides the effect of Hawking evaporation during the radiation dominated era.
The subsequent evaporation of the black holes in later eras is further modified
due to the variable gravitational ``constant'', and they could survive up to
longer times compared to the case of standard cosmology. We estimate the impact
of accretion on modification of the constraint on their initial mass fraction
obtained from the $\gamma$-ray background limit from presently evaporating
primordial black holes.
