Abstract: Poisson-Lie T-duality is a generalisation of traditional T-duality, allowing the nonlinear sigma model to be defined on target spaces without isometries. We apply the corresponding canonical transformations to bosonic open string worldsheet conformal boundary conditions, showing that the form of these conditions is invariant at the classical level; hence the dual model is again conformal. The boundary conditions are defined in terms of a gluing matrix which encodes the properties of D-branes, and we derive the duality map for this matrix. We demonstrate explicitly the implications of the duality map for D-branes in a simple non-Abelian Drinfel'd double.