It has been suggested that there could be a large number of primordial black holes which were formed in the early universe. We analyze the growth of such a primordial black hole following two different accretion rates - the Eddington accretion rate and the Bondi accretion rate - at the center of a host star like the sun. We find that a primordial black hole with M < ${sim}10^{17};g$ cannot substantially grow in any case throughout the lifetime of a host star. If M > ${sim}10^{17};g$, the evolution of a host star depends entirely on the mode of accretion, but it ends as a black hole in either case. Since more stars may have primordial black holes at the center of a galaxy this may result in a cluster of such black holes, and the cluster may eventually collapse to produce a single supermassive black hole.