We report on the first realization of a single bosonic Josephson junction, implemented by two weakly linked Bose-Einstein condensates in a double-well potential. In order to fully investigate the nonlinear tunneling dynamics we measure the density distribution in situ and deduce the relative phase between the two condensates from interference fringes. Our results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions. Additionally we confirm a novel nonlinear effect known as macroscopic quantum self-trapping, which leads to the inhibition of large amplitude tunneling oscillations.