The p-center problem consists in selecting p facilities from a set of possible sites and allocating a set of clients to them in such a way that the maximum distance between a client and the facility to which it is allocated is minimized. This paper proposes a new scalable exact solution algorithm based on client clustering and an iterative distance rounding procedure. The client clustering enables to initialize and update a subset of clients for which the p-center problem is iteratively solved. The rounding drastically reduces the number of distinct distances considered at each iteration. Our algorithm is tested on 396 benchmark instances with up to 1.9 million clients and facilities. We outperform the two state-of-the-art exact methods considered when p is not very small (i.e., p > 5).