  We put forward double perovskites of the R$_2$NiMnO$_6$ family (with $R$ a
rare-earth atom) as a new class of multiferroics on the basis of {\it ab
initio} density functional calculations. We show that changing $R$ from La to Y
drives the ground-state from ferromagnetic to antiferromagnetic with
$\uparrow\uparrow\downarrow\downarrow$ spin patterns. This E$^*$-type ordering
breaks inversion symmetry and generates a ferroelectric polarization of few
$\mu C/cm^2$. By analyzing a model Hamiltonian we understand the microscopic
origin of this transition and show that an external electric field can be used
to tune the transition, thus allowing electrical control of the magnetization.
