A new method is presented for measuring the spin of selectrons and smuons at the Large Hadron Collider (LHC), using an angular variable which is sensitive to the polar angle in direct slepton pair production. This variable, cos θ*_ll, is invariant under boosts along the beam axis, so it can be used at the LHC despite the fact that the longitudinal boost of the centre-of-mass frame cannot be determined. Monte Carlo simulations demonstrate that, using this method, the LHC can distinguish between the supersymmetric production angular distribution and phase space, or between supersymmetry and the production angular distribution of universal extra dimensions. An integrated luminosity of about 100 to 300 fb⁻¹ provides sufficient statistics to measure the slepton spin for points which had left-handed slepton masses in the range 202 to 338 GeV, and right-handed sleptons in the range 143 to 252 GeV. Good sensitivity was found in the `bulk' and `stau co-annihilation' regions of the cMSSM favoured by cosmological relic density measurements. Various systematic uncertainties are investigated, and some methods for reducing them are discussed.