  BiFeO$_3$ is the most famous multiferroic material, but it has no strong
spontaneous magnetization due to its antiferromagnetism. Here we show that
giant ferroelectric polarization and strong spontaneous magnetization can be
both realized in double perovskite Bi$_2$FeMoO$_6$ with R3 (\#146) space group
based on BiFeO$_3$. Our first-principles phonon spectra establishes that this
multiferroic R3 phase is stable. Our systematic calculations show that it is a
spin-polarized semiconductor with gap reaching to 0.54 eV and has a strong
ferroelectric polarization of 85$\mu$C/cm$^2$. This ferroelctricity is
comparable with that of BiFeO$_3$, but here obtained is a strong ferrimagnetism
with net magnetic moment of 2$\mu_B$ per formula unit and Curie temperature of
650 K. Both ferroelectric polarization and magnetic easy axis are shown to be
in pseudocubic [111] orientation. Our further analysis shows that the
macroscopic spontaneous magnetization can be deterministically reversed through
a three-step path by external electric field. Therefore, we believe that this
Bi$_2$FeMoO$_6$ material can be used to design new multifunctional materials
and achieve high-performance devices.
