  We simulate lattice QCD with dynamical $u$ and $d$ quarks at finite chemical
potential, $\mu_I$, for the third component of isospin ($I_3$), at both zero
and at finite temperature. At zero temperature there is some $\mu_I$, $\mu_c$
say, above which $I_3$ and parity are spontaneously broken by a charged pion
condensate. This is in qualitative agreement with the prediction of effective
(chiral) Lagrangians which also predict $\mu_c=m_\pi$. This transition appears
to be second order, with scaling properties consistent with the mean-field
predictions of such effective Lagrangian models. We have also studied the
restoration of $I_3$ symmetry at high temperature for $\mu_I > \mu_c$. For
$\mu_I$ sufficiently large, this finite temperature phase transition appears to
be first order. As $\mu_I$ is decreased it becomes second order connecting
continuously with the zero temperature transition.
