We present an algorithm for the transformation of a Gröbner basis of an ideal with respect to any given ordering into a Gröbner basis with respect to any other ordering. This algorithm is based on a modified version of the LLL algorithm. The worst case theoretical complexity of this algorithm is not better than the complexity of the FGLM algorithm; but can also give the theoretical complexity with some parameters depending on the size of the output. When the output is small then algorithm is more efficient. We also present a first implementation of the algorithm in Maple. This algorithm is restricted to the case of two variables but works also in positive dimension.