  Using canonical method we investigate $Dp-$brane world-volume
noncommutativity in weakly curved background. The term weakly curved means that
in the leading order, the source of non-flatness is infinitesimally small
Kalb-Ramond field $B_{\mu\nu}$ linear in coordinate, while the Ricci tensor
does not contribute being the infinitesimal of the second order. On the
solution of boundary conditions, we find simple expression for the space-time
coordinates in terms of the effective coordinates and momenta. This basic
relation helped us to prove that noncommutativity appears only on the
world-sheet boundary. The noncommutativity parameter has a standard form but
with infinitesimally small and coordinate dependent antisymmetric tensor
$B_{\mu\nu}$. This result coincides with that obtained on the group manifolds
in the limit of the large level $n$ of current algebra. After quantization the
algebra of functions on Dp-brane world-volume is represented with the
Kontsevich star product instead of the Moyal one in the flat background.
