  Bilayers consisting of two-dimensional (2D) electron and hole gases separated
by a 10 nm thick AlGaAs barrier are formed by charge accumulation in
epitaxially grown GaAs. Both vertical and lateral electric transport are
measured in the millikelvin temperature range. The conductivity between the
layers shows a sharp tunnel resonance at a density of $1.1 \cdot 10^{10} \text{
cm}^{-2}$, which is consistent with a Josephson-like enhanced tunnel
conductance. The tunnel resonance disappears with increasing densities and the
two 2D charge gases start to show 2D-Fermi-gas behavior. Interlayer
interactions persist causing a positive drag voltage that is very large at
small densities. The transition from the Josephson-like tunnel resonance to the
Fermi-gas behavior is interpreted as a phase transition from an exciton gas in
the Bose-Einstein-condensate state to a degenerate electron-hole Fermi gas.
