  We consider ground state of electron-hole graphene bilayer composed of two
independently doped graphene layers when a condensate of spatially separated
electron-hole pairs is formed. In the weak coupling regime the pairing affects
only conduction band of electron-doped layer and valence band of hole-doped
layer, thus the ground state is similar to ordinary BCS condensate. At strong
coupling, an ultrarelativistic character of electron dynamics reveals and the
bands which are remote from Fermi surfaces (valence band of electron-doped
layer and conduction band of hole-doped layer) are also affected by the
pairing. The analysis of instability of unpaired state shows that s-wave
pairing with band-diagonal condensate structure, described by two gaps, is
preferable. A relative phase of the gaps is fixed, however at weak coupling
this fixation diminishes allowing gapped and soliton-like excitations. The
coupled self-consistent gap equations for these two gaps are solved at zero
temperature in the constant-gap approximation and in the approximation of
separable potential. It is shown that, if characteristic width of the pairing
region is of the order of magnitude of chemical potential, then the value of
the gap in the spectrum is not much different from the BCS estimation. However,
if the pairing region is wider, then the gap value can be much larger and
depends exponentially on its energy width.
