  In the representation theory of finite groups, Brou\'e's abelian defect group
conjecture says that for any prime p if a p-block A of a finite group G has an
abelian defect group P, then A and its Brauer corresponding block B of the
normaliser N_G(P) of P in G are derived equivalent. We prove that Brou\'e's
conjecture, and even Rickard's splendid equivalence conjecture, are true for
the unique 3-block A of defect 2 of the sporadic simple Conway group Co_1,
implying that both conjectures hold for all 3-blocks of Co_1. To do so, we
determine the 3-decomposition numbers of A, and we actually show that A is Puig
equivalent to the principal 3-block of the symmetric group S_6 of degree 6.
