For a probability measure µ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is 1 as has been proven relatively recently. However, if µ 2N contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete mea- sures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by 1 for some constant 6 ≥ c > 2 cN which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation 1 is indeed N the optimal one.