We observed the starburst galaxy M82 in 850$mu$m polarised light with the POL-2 polarimeter on the James Clerk Maxwell Telescope (JCMT). We interpret our observed polarisation geometry as tracing a two-component magnetic field: a poloidal component aligned with the galactic superwind, extending to a height $sim 350$ pc above and below the central bar; and a spiral-arm-aligned, or possibly toroidal, component in the plane of the galaxy, which dominates the 850$mu$m polarised light distribution at galactocentric radii $gtrsim 2$ kpc. Comparison of our results with recent HAWC+ measurements of the field in the dust entrained by the M82 superwind suggests that the superwind breaks out from the central starburst at $sim 350$ pc above the plane of the galaxy.