We present a method to construct a patch of parametric surface of degree k+1 that fills a n-sided hole, with bigger than 2, and whose boundary coincides with a B-Spline, thus, the resulting patch can be easily connected with given B-Spline surfaces with fixed continuity conditions. The method is based on the generic approach by the same authors to con-struct free form surfaces, which gives a family of practical schemes to design surfaces from an arbitrary given mesh, using the differentiable manifold the-ory. The proposal uses a star shaped mesh which describes a generic n-hole and a surface in a neighborhood of the hole. From this mesh, a set of charts is defined, one associated to each vertex or face of the mesh, depending on the parity of the input parameter k. A basis function and a control point is defined from each chart, and the surface is obtained as a baricentric combination of the control points using the defined basis functions. The main advantages of the method are the following: arbitrary order k continuity conditions can be imposed; the involved hole can have an arbitrary number of sides and arbitrary shape (convex or not) the simplicity of the construction process gives an easy and flexible method; and finally, the surface near the boundary is a B-Spline with piecewise uniform knot sequences and whose control points are vertices of the given mesh. Implementation details to evaluate a surface point are given, showing that the de Boor algorithm can be exploited for efficiency.