  Filling factor $\nu=1$ bilayer electron systems in the quantum Hall regime
have an excitonic-condensate superfluid ground state when the layer separation
$d$ is less than a critical value $d_c$. On a quantum Hall plateau current
injected and removed through one of the two layers drives a dissipationless
edge current that carries parallel currents, and a dissipationless bulk
supercurrent that carries opposing currents in the two layers. In this paper we
discuss the theory of finite supercurrent bilayer states, both in the presence
and in the absence of symmetry breaking inter-layer hybridization. Solutions to
the microscopic mean-field equations exist at all condensate phase winding
rates for zero and sufficiently weak hybridization strengths. We find, however,
that collective instabilities occur when the supercurrent exceeds a critical
value determined primarily by a competition between direct and exchange
inter-layer Coulomb interactions. The critical current is estimated using a
local stability criterion and varies as $(d_c-d)^{1/2}$ when $d$ approaches
$d_c$ from below. For large inter-layer hybridization, we find that the
critical current is limited by a soliton instability of microscopic origin.
