  High-quality single crystals of NdFeAsO$_{0.65}$F$_{0.35}$ (the transition
temperature $T_{\rm c} \simeq 30.6$~K) were studied in zero-field (ZF) and
transverse-field (TF) muon-spin rotation/relaxation ($\mu$SR) experiments. An
upturn in muon-spin depolarization rate at $T\lesssim 3$~K was observed in
ZF-$\mu$SR measurements and it was associated with the onset of ordering of Nd
electronic moments. Measurements of the magnetic field penetration depth
($\lambda$) were performed in the TF geometry. By applying the external
magnetic field $B_{\rm ex}$ parallel to the crystallographic $c$-axis ($B_{\rm
ex}\| c$) and parallel to the $ab$-plane ($B_{\rm ex}\| ab$), the temperature
dependencies of the in-plane component ($\lambda_{ab}^{-2}$) and the
combination of the in-plane and the out of plane components
($\lambda_{ab,c}^{-2}$) of the superfluid density were determined,
respectively. The out-of-plane superfluid density component
($\lambda_{c}^{-2}$) was further obtained by combining the results of $B_{\rm
ex} \| c$ and $B_{\rm ex} \| {ab}$ set of experiments. The temperature
dependencies of $\lambda_{ab}^{-2}$, $\lambda_{ab,c}^{-2}$, and
$\lambda_{c}^{-2}$ were analyzed within the framework of a self-consistent
two-gap model despite of using the traditional $\alpha$-model. Interband
coupling was taken into account, instead of assuming it to be zero as it stated
in the $\alpha$-model. A relatively small value of the interband coupling
constant $\Lambda_{12} \simeq 0.01$ was obtained, thus indicating that the
energy bands in NdFeAsO$_{0.65}$F$_{0.35}$ are only weakly coupled. In spite of
their small magnitude, the coupling between the bands leads to the single value
of the superconducting transition temperature $T_{\rm c}$.
