  We report detailed magnetization ($M$) and magnetoresistance ($MR$) studies
on a series of Sr$_2$FeMoO$_6$ samples with independent control on anti-site
defect and grain boundary densities. These results, exhibiting a switching-like
behavior of $MR$ with $M$, establish that the $MR$ is controlled by the
magnetic polarization of grain boundary regions, rather than of the grains
within a resonant tunnelling mechanism.
