  The intermittency of a passive scalar advected by three-dimensional
Navier-Stokes turbulence at a Taylor-scale Reynolds number of $650$ is studied
using direct numerical simulations on a $4096^3$ grid; the Schmidt number is
unity. By measuring scalar increment moments of high orders, while ensuring
statistical convergence, we provide unambiguous evidence that the scaling
exponents saturate to $1.2$ for moment order beyond about $12$, indicating that
scalar intermittency is dominated by the most singular shock-like cliffs in the
scalar field. We show that the fractal dimension of the spatial support of
steep cliffs is about $1.8$, whose sum with the saturation exponent value of
$1.2$ adds up to the space dimension of $3$, thus demonstrating a deep
connection between the geometry and statistics in turbulent scalar mixing. The
anomaly for the fourth and sixth order moments is comparable to that in the
Kraichnan model for the roughness exponent of $2/3$.
