  We begin the study of a tilting theory in certain truncated categories of
modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a
finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$,
$J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral
weights of the simple Lie algebra. We use this to put a tilting theory on the
category $\mathcal G(\Gamma')$ where $\Gamma' = P' \times J$, where
$P'\subseteq P^+$ is saturated. Under certain natural conditions on $\Gamma'$,
we note that $\mathcal G(\Gamma')$ admits full tilting modules.
