The invariant-mass spectrum of e+e- pairs produced in 12C+12C collisions at an incident energy of 2 GeV per nucleon has been measured for the first time. The measured pair production probabilities span over 5 orders of magnitude from the pi(0)-Dalitz to the rho/omega invariant-mass region. Dalitz decays of pi(0) and eta account for all the yield up to 0.15 GeV/c(2), but for only about 50% above this mass. A comparison with model calculations shows that the excess pair yield is likely due to baryon-resonance and vector-meson decays. Transport calculations based on vacuum spectral functions fail, however, to describe the entire mass region.