  We consider the classical optimal dividends problem under the
Cram\'er-Lundberg model with exponential claim sizes subject to a constraint on
the time of ruin. We introduce the dual problem and show that the complementary
slackness conditions are satisfied, thus there is no duality gap. Therefore the
optimal value function can be obtained as the point-wise infimum of auxiliary
value functions indexed by Lagrange multipliers. We also present a series of
numerical examples.
