We investigate the evolution of the close binary system consisting of
the millisecond radio pulsar PSR J1012+53 and its low-mass companion,
presumably a helium white dwarf. Within a standard evolutionary scenar
io, we explore the most plausible range of masses each component would
have when the orbital parameters of the system achieve the observed v
alues. We demonstrate that a neutron star of initial mass 1.4 M. can a
ccrete, at most, 0.75 M. during the whole evolution of this binary sys
tem and may have increased its gravitational mass to 2.0 M.. Also, our
evolutionary calculations reveal that, for all relevant scenarios, th
e lowest mass the helium white dwarf in this system can reach at the l
atest stages of binary evolution is 0.15 M.. We present estimates of t
he age of the neutron star in this system that consistently account fo
r all phases of detached and semidetached binary evolution, including
the rather prolonged phase of a white dwarf cooling. For the system PS
R J1012+53, we demonstrate the occurrence of thermal hydrogen flashes
on a low-mass helium white dwarf at the latest stages of its evolution
. We discuss the impact of this system on the issue of the mass limit
for a neutron star and on the problem of the structure and evolution o
f very low mass helium white dwarfs, companions of millisecond pulsars
.