Well, I've to look that issue up again, but the microcanonical ensemble is usually defined as follows.
Let |n \rangle be a complete set of orthonormal energy eigenvectors. Then the microcanonical ensemble is defined by
\hat{R}=\frac{1}{\omega(E,\Delta E)} \sum_{n, E_n \in (E-\Delta E,E)} |n \rangle \langle n|
with
\omega(E,\Delta E)=\sum_{n,E_n \in (E-\Delta E,E)}.
This is the maximum entropy under the constraint that the energy of the system is strictly in an interval (E-\Delta E,E).
The entropy is given by the von Neumann expression
S_{\text{MC}}=-\mathrm{Tr} \; \hat{R} \ln \hat{R}=\ln \omega(E,\Delta E).
The definition of the entropy by Dunkel an Hilbert also refers to the von Neuman entropy maximized under the constraint that the energy is strictly below the value E, i.e., the Statistical operator is given by
\hat{R}_{\text{DH}}=\frac{1}{\Omega(E)} \sum_{n, E_n \leq E} |n \rangle \langle n| = \sum_{n} \Theta(E-E_n) |n \rangle \langle n|
with
\Omega(E)=\sum_n \Theta(E-E_n), \quad S_{\text{DH}}=\ln \Omega(E).
Then of course, the temperature is strictly positive, if defined by the thermodynamic relation
\beta=\frac{1}{T} = \frac{\partial S}{\partial E},
where all the external parameters (like the volume of a container of a gas) are kept constant when taking the derivative, i.e., for a gas S=S(E,V). In the thermodynamic limit the usual definition of the of the microcanonical and the Dunkel-Hilbert definition of the thermodynamic quantities coincide asymptotically for large systems.
BTW: I guess you refer to the arXiv version of the paper, which is a bit more detailed than the published version in Nature.
I've just found the following inteteresting reference about this issue, where this is explained in much more detail:
R. B. Griffiths, Microcanonical Ensemble in Quantum Statistical Mechanics, Journal of Mathematical Physics
6, 1447
http://dx.doi.org/10.1063/1.1704681