Recently Andersson et. al., and Bildsten have independently suggested that an r-mode instability might be responsible for stalling the neutron-star spin-up in strongly accreting, Low Mass X-ray Binaries (LMXBs). We show that if this does occur, then there are two possibilities for the resulting neutron-star evolution: If the r-mode damping is a decreasing function of temperature, then the star undergoes a cyclic evolution: (i) accretional spin-up triggers the instability near the observed maximum spin rate; (ii) the r-modes become highly excited through gravitational-radiation reaction, and in a fraction of a year they viscously heat the star; (iii) r-mode gravitational-radiation reaction then spins the star down in a fraction of a year to some limiting rotational frequency; (iv) the r-mode instability shuts off; (v) the neutron star slowly cools and is spun up by accretion, until it once again reaches the instability point, closing the cycle. The shortness of the epoch of r-mode activity makes it unlikely that r-modes are currently excited in the neutron star of any galactic LMXBs. Nevertheless, this cyclic evolution could be responsible for keeping the rotational frequencies within the observed LMXB frequency range. If, on the other hand, the r-mode damping is temperature independent, then a steady state with constant angular velocity and $T_{\rm core}\simeq 4\times 10^8$K is reached, in which r-mode viscous heating is balanced by neutrino cooling and accretional spin-up torque is balanced by gravitational-radiation-reaction spin-down torque. In this case the neutron stars in LMXBs could be potential sources of periodic gravitational waves, detectable by enhanced LIGO interferometers.

20 pages, 1 ps figure