  In the present paper we prove that the classes of Dixmier and Connes-Dixmier
traces differ even on the Dixmier ideal $\mathcal M_{1,\infty}$. We construct a
Marcinkiewicz space $\mathcal M_\psi$ and a positive operator $T\in \mathcal
M_\psi$ which is Connes-Dixmier measurable but which is not Dixmier measurable.
