  We characterize the spectrum of one-dimensional Schr\"odinger operators
H=-d^2/dx^2+V with quasi-periodic complex-valued algebro-geometric potentials V
(i.e., potentials V which satisfy one (and hence infinitely many) equation(s)
of the stationary Korteweg-de Vries (KdV) hierarchy) associated with
nonsingular hyperelliptic curves. The corresponding problem appears to have
been open since the mid-seventies. The spectrum of H coincides with the
conditional stability set of H and can explicitly be described in terms of the
mean value of the inverse of the diagonal Green's function of H.
  As a result, the spectrum of H consists of finitely many simple analytic arcs
and one semi-infinite simple analytic arc in the complex plane. Crossings as
well as confluences of spectral arcs are possible and discussed as well.
