Polarisation asymmetries are measured for the hard exclusive leptoproduction of real photons from a longitudinally polarised hydrogen target. These asymmetries arise from the deeply virtual Compton scattering and Bethe-Heitler processes. From the data are extracted two asymmetries in the azimuthal distribution of produced real photons about the direction of the exchanged virtual photon: A_UL with respect to the target polarisation and A_LL with respect to the product of the beam and target polarisations. Results for both asymmetries are compared to the predictions from a generalised parton distribution model. The sin(phi) and cos(0*phi) amplitudes observed respectively for the A_UL and A_LL asymmetries are compatible with the sizeable predictions from the model. Unexpectedly, a sin(2*phi) modulation in the A_UL asymmetry with a magnitude similar to that of the sin(phi) modulation is observed.