  We construct the law of L\'{e}vy processes conditioned to stay positive under
general hypotheses. We obtain a Williams type path decomposition at the minimum
of these processes. This result is then applied to prove the weak convergence
of the law of L\'{e}vy processes conditioned to stay positive as their initial
state tends to 0. We describe an absolute continuity relationship between the
limit law and the measure of the excursions away from 0 of the underlying
L\'{e}vy process reflected at its minimum. Then, when the L\'{e}vy process
creeps upwards, we study the lower tail at 0 of the law of the height this
excursion.
