A Poisson process P-lambda on R-d with causal structure inherited from the the usual Minkowski metric on R-d has a normalised discrete causal distance D-lambda (x, y) given by the height of the longest causal chain normalised by lambda(1/d)c(d). We prove that P-lambda restricted to a compact set Q converges in probability in the sense of Noldus (2004 Class. Quantum Grav. 21 839-50) to Q with the Minkowski metric.