  A word equation with one variable in a free group is given as $U = V$, where
both $U$ and $V$ are words over the alphabet of generators of the free group
and $X, X^{-1}$, for a fixed variable $X$. An element of the free group is a
solution when substituting it for $X$ yields a true equality (interpreted in
the free group) of left- and right-hand sides. It is known that the set of all
solutions of a given word equation with one variable is a finite union of sets
of the form $\{\alpha w^i \beta \: : \: i \in \mathbb Z \}$, where $\alpha, w,
\beta$ are reduced words over the alphabet of generators, and a polynomial-time
algorithm (of a high degree) computing this set is known. We provide a cubic
time algorithm for this problem, which also shows that the set of solutions
consists of at most a quadratic number of the above-mentioned sets. The
algorithm uses only simple tools of word combinatorics and group theory and is
simple to state. Its analysis is involved and focuses on the combinatorics of
occurrences of powers of a word within a larger word.
