  In neutral cold quark matter that is sufficiently dense that the strange
quark mass M_s is unimportant, all nine quarks (three colors; three flavors)
pair in a color-flavor locked (CFL) pattern, and all fermionic quasiparticles
have a gap. We argue that as a function of decreasing quark chemical potential
mu or increasing M_s, there is a quantum phase transition from the CFL phase to
a new ``gapless CFL phase'' in which only seven quasiparticles have a gap. The
transition occurs where M_s^2/mu is approximately equal to 2*Delta, with Delta
the gap parameter. Gapless CFL, like CFL, leaves unbroken a linear combination
Qtilde of electric and color charges, but it is a Qtilde-conductor with a
nonzero electron density. These electrons and the gapless quark quasiparticles
make the low energy effective theory of the gapless CFL phase and,
consequently, its astrophysical properties are qualitatively different from
those of the CFL phase, even though its U(1) symmetries are the same. Both
gapless quasiparticles have quadratic dispersion relations at the quantum
critical point. For values of M_s^2/mu above the quantum critical point, one
branch has conventional linear dispersion relations while the other branch
remains quadratic, up to tiny corrections.
