  In this paper we consider the quantization of open strings ending on D-branes
with a background B field. We find that spacetime coordinates of the open
string end-points become noncommutative, and correspondingly the D-brane
worldvolume also becomes noncommutative. This provides a string theory
derivation and generalization of the noncommutativity obtained previously in
the Matrix model compactification. For Dp-branes with p>=2 our results are new
and agree with that of Matrix theory for the case of A=0 (where $A$ is the
worldvolume gauge field) if the T-duality radii are used.
