We present observations taken with SCUBA on the JCMT of the Horsehead Nebula in Orion (B33), at wavelengths of 450 and 850 \mum. We see bright emission from that part of the cloud associated with the photon-dominated region (PDR) at the `top' of the horse's head, which we label B33-SMM1. We characterise the physical parameters of the extended dust responsible for this emission, and find that B33-SMM1 contains a more dense core than was previously suspected. We compare the SCUBA data with data from the Infrared Space Observatory (ISO) and find that the emission at 6.75-\mum is offset towards the west, indicating that the mid-infrared emission is tracing the PDR while the submillimetre emission comes from the molecular cloud core behind the PDR. We calculate the virial balance of this core and find that it is not gravitationally bound but is being confined by the external pressure from the HII region IC434, and that it will either be destroyed by the ionising radiation, or else may undergo triggered star formation. Furthermore we find evidence for a lozenge-shaped clump in the `throat' of the horse, which is not seen in emission at shorter wavelengths. We label this source B33-SMM2 and find that it is brighter at submillimetre wavelengths than B33-SMM1. SMM2 is seen in absorption in the 6.75-\mum ISO data, from which we obtain an independent estimate of the column density in excellent agreement with that calculated from the submillimetre emission. We calculate the stability of this core against collapse and find that it is in approximate gravitational virial equilibrium. This is consistent with it being a pre-existing core in B33, possibly pre-stellar in nature, but that it may also eventually undergo collapse under the effects of the HII region.Read less <