  We give a necessary and sufficient condition on a positive compact operator
$T$ for the existence of a singular trace (i.e. a trace vanishing on the finite
rank operators) which takes a finite non-zero value on $T$. This generalizes
previous results by Dixmier and Varga. We also give an explicit description of
these traces and associated ergodic states on $\ell^\infty(\Bbb N)$ using tools
of non standard analysis in an essential way.
