A $p$-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined $p$-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for $p$ an odd prime, proving that there is a one-to-one correspondence between connected $p$-compact groups and finite reflection groups over the $p$-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as $p$-compact groups by their Weyl groups seen as finite reflection groups over the $p$-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for $p$ odd.