  A (noncommutative) P\'olya series over a field $K$ is a formal power series
whose nonzero coefficients are contained in a finitely generated subgroup of
$K^\times$. We show that rational P\'olya series are unambiguous rational
series, proving a 40 year old conjecture of Reutenauer. The proof combines
methods from noncommutative algebra, automata theory, and number theory
(specifically, unit equations). As a corollary, a rational series is a P\'olya
series if and only if it is Hadamard sub-invertible. Phrased differently, we
show that every weighted finite automaton taking values in a finitely generated
subgroup of a field (and zero) is equivalent to an unambiguous weighted finite
automaton.
