  We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2$) with
fractional diffusions $\Lambda^{2\alpha}\boldsymbol{u}$ and
$\Lambda^{2\beta}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the
magnetic field $\boldsymbol{b}$, respectively. Here $\alpha,\beta$ are positive
constants and $\Lambda^s = (-\Delta)^{s/2}$ is the fractional Laplacian of
order $s$. We establish global existence of weak solutions of the
Stokes-Magneto system for any initial data in $L_{2}$ when $\alpha$, $\beta$
satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$, and
$\min\{\alpha+\beta,2\alpha+\beta-1\}>d/2$. It is also shown that weak
solutions are unique if $\beta \geq 1$ and $\min
\{\alpha+\beta,2\alpha+\beta-1\}\geq d/2+1$, in addition.
