In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter θ. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with ∆n = sup i=0,...,n−1 (t n i+1 − t n i) → 0. We introduce an estimator of θ, based on a contrast function, which is efficient without requiring any conditions on the rate at which ∆n → 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n∆ 3 n → 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of θ is feasible under the condition that n∆ k n → 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. Lévy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.