  Intermittency is investigated using decaying direct numerical simulations of
incompressible weak magnetohydrodynamic turbulence with a strong uniform
magnetic field ${\bf b_0}$ and zero cross-helicity. At leading order, this
regime is achieved via three-wave resonant interactions with the scattering of
two of these waves on the third/slow mode for which $k_{\parallel} = 0$. When
the interactions with the slow mode are artificially reduced the system
exhibits an energy spectrum with $k_{\perp}^{-3/2}$, whereas the expected exact
solution with $k_{\perp}^{-2}$ is recovered with the full nonlinear system. In
the latter case, strong intermittency is found when the vector separation of
structure functions is taken transverse to ${\bf b_0}$ - at odds with classical
weak turbulence where self-similarity is expected. This surprising result,
which is being reported here for the first time, may be explained by the
influence of slow modes whose regime belongs to strong turbulence. We derive a
new log--Poisson law, $\zeta_p = p/8 +1 -(1/4)^{p/2}$, which fits perfectly the
data and highlights the dominant role of current sheets.
