  Tight binaries of helium white dwarfs (He WDs) orbiting millisecond pulsars
(MSPs) will eventually "merge" due to gravitational damping of the orbit. The
outcome has been predicted to be the production of long-lived ultra-compact
X-ray binaries (UCXBs), in which the WD transfers material to the accreting
neutron star (NS). Here we present complete numerical computations, for the
first time, of such stable mass transfer from a He WD to a NS. We have
calculated a number of complete binary stellar evolution tracks, starting from
pre-LMXB systems, and evolved these to detached MSP+WD systems and further on
to UCXBs. The minimum orbital period is found to be as short as 5.6 minutes. We
followed the subsequent widening of the systems until the donor stars become
planets with a mass of ~0.005 Msun after roughly a Hubble time. Our models are
able to explain the properties of observed UCXBs with high helium abundances
and we can identify these sources on the ascending or descending branch in a
diagram displaying mass-transfer rate vs. orbital period.
