  We study the quantum geometry of the fuzzy sphere defined as the angular
momentum algebra $[x_i,x_j]=2\imath\lambda_p \epsilon_{ijk}x_k$ modulo setting
$\sum_i x_i^2$ to a constant, using a recently introduced 3D rotationally
invariant differential structure. Metrics are given by symmetric $3 \times 3$
matrices $g$ and we show that for each metric there is a unique quantum
Levi-Civita connection with constant coefficients, with scalar curvature $
\frac{1}{2}({\rm Tr}(g^2)-\frac{1}{2}{\rm Tr}(g)^2)/\det(g)$. As an
application, we construct Euclidean quantum gravity on the fuzzy unit sphere.
We also calculate the charge 1 monopole for the 3D differential structure.
