  We show that any normal toric variety over a rank one valuation ring admits
an equivariant open embedding in a normal toric variety which is proper over
the valuation ring, after a base-change by a finite extension of valuation
rings. If the value group $\Gamma$ is discrete or divisible then no base-change
is needed. We give explicit examples which show that existing methods do not
produce such normal equivariant completions. Our approach is combinatorial and
proceeds by showing that $\Gamma$-admissible fans admit $\Gamma$-admissible
completions. In order to show this we prove a combinatorial analog of
noetherian reduction which we believe will be of independent interest.
