  We exhibit an action of Conway's group---the automorphism group of the Leech
lattice---on a distinguished super vertex operator algebra, and we prove that
the associated graded trace functions are normalized principal moduli, all
having vanishing constant terms in their Fourier expansion. Thus we construct
the natural analogue of the Frenkel--Lepowsky--Meurman moonshine module for
Conway's group.
  The super vertex operator algebra we consider admits a natural
characterization, in direct analogy with that conjectured to hold for the
moonshine module vertex operator algebra. It also admits a unique
canonically-twisted module, and the action of the Conway group naturally
extends. We prove a special case of generalized moonshine for the Conway group,
by showing that the graded trace functions arising from its action on the
canonically-twisted module are constant in the case of Leech lattice
automorphisms with fixed points, and are principal moduli for genus zero groups
otherwise.
