  This paper studies the class of unique factorial domains $B$ over an
algebraically closed field $k$ which are affine or unirational over $k$ and
which admit an effective unmixed $\mathbb{Z}^{d-1}$-grading with $B_0=k$, where
$d$ is the dimension of $B$. Geometrically, these correspond to factorial
affine $k$-varieties with an unmixed torus action of complexity one and trivial
invariants. Our main result shows that this class is identical to the class of
rings defined by trinomial data, thus generalizing earlier work of Mori, of
Ishida, and of Hausen, Herrppich and S\"uss.
