An algorithm is presented that constructs a C/sup 2/-continuous B-spline quaternion curve which interpolates a given sequence of unit quaternions on the rotation group SO(3). The de Casteljau type construction method of B-spline curves can be extended to generate B-spline quaternion curves; however, the B-spline quaternion curves do not have C/sup 2/-continuity in SO(3). The authors recently suggested a new construction method that can extend a B-spline curve to a similar one in SO(3) while preserving the C/sup k/-continuity of the B-spline curve. We adapt this method for the construction of a B-spline quaternion interpolation curve. Thus, the problem essentially reduces to the problem of finding the control points for the B-spline interpolation curve. However, due to the non-linearity of the associated constraint equations, it is non-trivial to compute the B-spline control points. We provide an efficient iterative refinement solution which can approximate the control points very precisely.