In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups BS(1,k), k>1, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulae for the series. As a result of our computation we also establish that in each BS(1,k) the conjugacy and standard growth rates are equal.

The authors acknowledge the hospitality of the Hausdorff Institute in Bonn and thank the organisers of the Trimester Logic and Algorithms in Group Theory, where the discussions on this paper started. They would also like to thank the referee for their helpful comments. The first named author was partially supported by EPSRC Standard grant EP/R035814/1.