  The invariant-mass spectrum of $e^+ e^-$ pairs produced in $^{12}$C+$^{12}$C
collisions at an incident energy of 2 GeV per nucleon has been measured for the
first time. The measured pair production probabilities span over five 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.
