  We prove that if an alternating knot has unknotting number one, then there
exists an unknotting crossing in any alternating diagram. This is done by
showing that the obstruction to unknotting number one developed by Greene in
his work on alternating 3-braid knots is sufficient to identify all unknotting
number one alternating knots. As a consequence, we also get a converse to the
Montesinos trick: an alternating knot has unknotting number one if its branched
double cover arises as half-integer surgery on a knot in $S^3$. We also reprove
a characterisation of almost-alternating diagrams of the unknot originally due
to Tsukamoto.
