We provide a complete classification of the extreme rays of the $6 \times 6$ copositive cone ${\cal COP}^6$. We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $A \in {\cal COP}^6$. To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric $6 \times 6$ matrices ${\cal S}^6$, parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices $A \in {\cal COP}^6$ having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros $u$ as well as the supports of the corresponding matrix-vector products $Au$. The analysis uses recently and newly developed methods that are applicable also to copositive matrices of arbitrary order.