Alternating knots with unknotting number one

We show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram. We also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half-integer surgery on a knot in S3, thus establishing a converse to the Montesinos trick. Along the way, we reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto. These results are established using the obstruction to unknotting number one developed by Greene.