  We study the evolution of accreting oxygen-neon (ONe) white dwarfs (WDs),
with a particular emphasis on the effects of the presence of the carbon-burning
products $\mathrm{^{23}Na}$ and $\mathrm{^{25}Mg}$. These isotopes lead to
substantial cooling of the WD via the $\mathrm{^{25}Mg}$-$\mathrm{^{25}Na}$,
$\mathrm{^{23}Na}$-$\mathrm{^{23}Ne}$, and
$\mathrm{^{25}Na}$-$\mathrm{^{25}Ne}$ Urca pairs. We derive an analytic formula
for the peak Urca-process cooling rate and use it to obtain a simple expression
for the temperature to which the Urca process cools the WD. Our estimates are
equally applicable to accreting carbon-oxygen WDs. We use the Modules for
Experiments in Stellar Astrophysics (MESA) stellar evolution code to evolve a
suite of models that confirm these analytic results and demonstrate that
Urca-process cooling substantially modifies the thermal evolution of accreting
ONe WDs. Most importantly, we show that MESA models with lower temperatures at
the onset of the $\mathrm{^{24}Mg}$ and $\mathrm{^{24}Na}$ electron captures
develop convectively unstable regions, even when using the Ledoux criterion. We
discuss the difficulties that we encounter in modeling these convective regions
and outline the potential effects of this convection on the subsequent WD
evolution. For models in which we do not allow convection to operate, we find
that oxygen ignites around a density of $\log(\rho_{\rm c}/\rm g\,cm^{-3})
\approx 9.95$, very similar to the value without Urca cooling. Nonetheless, the
inclusion of the effects of Urca-process cooling is an important step in
producing progenitor models with more realistic temperature and composition
profiles which are needed for the evolution of the subsequent oxygen
deflagration and hence for studies of the signature of accretion-induced
collapse.
