Countless algorithms have been developed for the multiplication of univariate polynomials and multi-precision integers, but all those with sub-quadratic time complexity currently require at least Ω(n) extra space for the computation. A new routine based on the Karatsuba/Ofman algorithm is presented with the same time complexity of O(n^1.59) but only O(log n) extra space. A second routine based on the method of Schönhage/Strassen achieves the same pseudo-linear time and O(1) extra space, but only under certain conditions. A preliminary implementation over Z/Z_p[x], where p fits into a single machine word, is presented and compared with existing software.