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We consider positive solutions of the system $u_t-\Delta u=v^p$; $v_t-\Delta v=u^q$ in a ball or in the whole space, with $p,q>1$. Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p=q$. Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for the final...