Most Quantum Key Distribution protocols use a two-dimensional basis such as HV polarization as first proposed by Bennett and Brassard in 1984. These protocols are consequently limited to a key generation density of 1 bit per photon. We increase this key density by encoding information in the transverse spatial displacement of the used photons. Employing this higher-dimensional Hilbert space together with modern single-photon-detecting cameras, we demonstrate a proof-of-principle large-alphabet Quantum Key Distribution experiment with 1024 symbols and a shared information between sender and receiver of 7 bit per photon.