  We examine the phase structure of massive Yang-Mills theory in 1+1
dimensions. This theory is equivalent to a gauged principal chiral sigma model.
It has been previously shown that the gauged theory has only a confined phase,
and no Higgs phase in the continuum, and at infinite volume. There are no
massive gluons, but only hadron-like bound states of sigma-model particles. The
reason is that the gluon mass diverges, being proportional to the two-point
correlation function of the renormalized field of the sigma model at $x=0$. We
use exact large-$N$ results to show that after introducing a lattice
regularization and typical values of the coupling constants used in Monte Carlo
simulations, the gluon mass becomes finite, and even sometimes small. A smooth
crossover into a Higgs phase can then appear. For small volumes and large $N$,
we find an analytic expression for the gluon mass, which depends on the
coupling constants and the volume. We argue that this Higgs phase is
qualitatively similar to the one observed in lattice computations at $N=2$.
