  We prove that the mapping class group $\Gamma_{g,n}$ for surfaces of negative
Euler characteristic has a cofinite universal space $\E$ for proper actions
(the resulting quotient is a finite $CW$-complex). The approach is to construct
a truncated Teichmueller space $\T_{g,n}(\epsilon)$ by introducing a lower
bound for the length of shortest closed geodesics and showing that
$\T_{g,n}(\epsilon)$ is a $\Gamma_{g,n}$ equivariant deformation retract of the
Teichmueller space $\T_{g, n}$. The existence of such a cofinite universal
space is important in the study of the cohomology of the group $\gag$. As an
application, we note that there are only finitely many conjugacy classes of
finite subgroups of $\Gamma_{g,n}$. Another application is that the rational
Novikov conjecture in K-theory holds for $\Gamma_{g,n}$.
