  Super-Eddington accretion is very efficient in growing the mass of a black
hole: in a fraction of the Eddington time its mass can grow to an arbitrary
large value if the feedback effect is not taken into account. However, since
super-Eddington accretion has a very low radiation efficiency, people have
argued against it as a major process for the growth of the black holes in
quasars since observations have constrained the average accretion efficiency of
the black holes in quasars to be $\ga 0.1$. In this paper we show that the
observational constraint does not need to be violated if the black holes in
quasars have undergone a two-phase growing process: with a short
super-Eddington accretion process they get their masses inflated by a very
large factor until the feedback process becomes important, then with a
prolonged sub-Eddington accretion process they have their masses increased by a
factor $\ga 2$. The overall average efficiency of this two-phase process is
then $\ga 0.1$, and the existence of black holes of $10^9 M_\odot$ by redshift
6 is easily explained. Observational test of the existence of the
super-Eddington accretion phase is briefly discussed.
