Millisecond pulsars (MSPs) are old, rapidly rotating neutron stars with low spin-down rates. Their properties point towards an origin in low-mass X-ray binaries (LMXBs), in which mass transfer onto the neutron stars makes them spin up to their observed, fast rotation. The lack of objects rotating near break-up and the similar rotation periods of several LMXBs have been attributed to the accretion torque being balanced, at fast rotation, by gravitational radiation (Wagoner 1984, Bildsten 1998), perhaps caused by an r-mode made unstable through the so-called Chandrasekhar-Friedman-Schutz (CFS) mechanism. Internal dissipation through weak interaction processes involving Lambda^0 and Sigma^- hyperons has been argued to cause LMXBs to evolve into a quasi-steady state, in which the neutron star has a nearly constant rotation rate, temperature, and mode amplitude (Wagoner 2002). We take this hypothesis one step further, showing that MSPs descending from these LMXBs spend a long time in a similar state, in which they are extremely steady sources of both gravitational radiation and X-rays, possibly detectable by existing or planned instruments. We also show that the dissipation consistent with the spin-down of observed MSPs is very strong. The same dissipation, applied to LMXBs, prevents their oscillation modes from reaching an amplitude large enough to limit the rotation of the neutron star to frequencies much smaller than the mass-shedding limit.