  In AM CVn binaries, a white dwarf primary accretes material from a
helium-rich white dwarf or stellar companion. The unstable ignition of nuclear
burning via the $3\alpha$ reaction in an accumulated helium layer powers a
thermonuclear runaway near accretion rates $\dot{M} \lesssim 10^{-6} \,
\mathrm{M_{\odot} \ yr^{-1}}$ that may be observed as helium nova or .Ia
supernova. Helium burning in the primary's envelope at temperatures $T \lesssim
10^{8} \, \mathrm{K}$ may proceed via the direct three-body fusion of
$\alpha$-particles. Here we show that the direct three-body rate by Nguyen et
al. (2012) -- which is reduced relative to the extrapolated resonant rate at
temperatures $T \gtrsim 5 \times 10^{7} \, \mathrm{K}$ -- results in novae with
longer recurrence times and larger ignition masses. By contrast, we find that
the enhancement in the direct three-body rate at temperatures below $T \lesssim
5 \times 10^{7} \, \mathrm{K}$ does not result in significant differences in
nova outburst properties. The most massive envelopes in our models are near the
density threshold for detonation of the helium layer, where an increase in the
density at ignition due to the $3\alpha$ rate may be important.
