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.