  A discretization of the peakons lattice is introduced, belonging to the same
hierarchy as the continuous--time system. The construction examplifies the
general scheme for integrable discretization of systems on Lie algebras with
$r$--matrix Poisson brackets. An initial value problem for the difference
equations is solved in terms of a factorization problem in a group.
Interpolating Hamiltonian flow is found. A variational (Lagrangian) formulation
is also given.
