We propose an algorithm for the construction of a nearly optimal integer to integer approximation of the Karhunen-Loeve Transform. The algorithm is based on the method of P. Hao and Q. Shi as described in [1] but-unlike described in the paper-we vary the pivoting in order to obtain a better approximation of the linear transform. We have then developed an algorithm for hyperspectral image lossless compression that uses first an Integer Wavelet Transform for the spatial decorrelation, then our Integer-KLT for the spectral decorrelation, and finally a 3D context-based adaptive arithmetic coding (3D-CBAC), which exploits the dependencies among symbols. The results on the AVIRIS images are better than those indicated in literature.