  In this paper, we continue the construction of variational integrators
adapted to contact geometry started in \cite{VBS}, in particular, we introduce
a discrete Herglotz Principle and the corresponding discrete Herglotz Equations
for a discrete Lagrangian in the contact setting. This allows us to develop
convenient numerical integrators for contact Lagrangian systems that are
conformally contact by construction. The existence of an exact Lagrangian
function is also discussed.
