Complexity of two-variable logic with counting

Let C/sub k//sup 2/ denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) m", for m/spl les/k, and let C/sup 2/ be the union of C/sub k//sup 2/ taken over all integers k. We prove that the problem of satisfiability of sentences of C/sub 1//sup 2/ is NEXPTIME-complete. This strengthens a recent result of E. Gradel, Ph. Kolaitis and M. Vardi (1997) who proved that the satisfiability problem for the first order two-variable logic L/sup 2/ is NEXPTIME-complete and a very recent result by E. Gradel, M. Otto and E. Rosen (1997) who proved the decidability of C/sup 2/. Our result easily implies that the satisfiability problem for C/sup 2/ is in non-deterministic, doubly exponential time. It is interesting that C/sub 1//sup 2/ is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size.