We observe the total filling factor νT=1 exciton condensate in independently contacted bilayer two-dimensional electron systems in samples with minute tunnel coupling. At balanced electron densities in the layers, we find for both drag and counter-flow current configurations, thermally activated transport with a monotonic increase of the activation energy for d/ℓB<1.65 with activation energies up to 0.4 K. In the imbalanced system the activation energies show a striking asymmetry around the balance point, implying that the gap to charge excitations is considerably different in the separate layers that form the bilayer condensate. This indicates that the measured activation energy is neither the binding energy of the excitons, nor their condensation energy.