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 $Δ$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $θ$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $Δ$n $\rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$Δ$ 3 n $\rightarrow$ 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$Δ$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{é}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.