  This paper presents multi-filter measurements of the night sky brightness at
the South African Astronomical Observatory (SAAO) in Sutherland in the presence
of a bright moon. The observations cover a wide range of sky directions, lunar
phases and lunar positions. A revised simplified scattering model is developed
for estimating the sky brightness due to moonlight that more accurately
reflects the atmospheric extinction of the lunar beam compared to models
frequently applied in astronomical studies. Contributions to night sky
brightness due to sources other than moonlight are quantified and subtracted
from the total sky background radiation to determine the spectral intensity and
angular distribution of scattered moonlight. The atmospheric scattering phase
function is then derived by comparing the sky brightening to the strength of
the incoming lunar beam, estimated using a novel approach. The phase function
is shown to be an excellent match to the combined theoretical Rayleigh and Mie
scattering functions, the latter with a Henyey--Greenstein form instead of the
exponential angular relationship often used in previous studies. Where
deviations between measured and model sky brightness are evident in some bands
these are explained by contributions from multiple scattering or airglow, and
are quantified accordingly. The model constitutes an effective tool to predict
sky brightness at SAAO in optical photometric bands, especially with a bright
moon present. The methodology can also be readily be adapted for use at other
astronomical sites. The paper furthermore presents $UBV(RI)_c$ and
Str{\"o}mgren photometry for 49 stars, most with no prior such data.
