  Supermassive stars (SMSs) forming via very rapid mass accretion (Mdot >~ 0.1
Msun/yr) could be precursors of supermassive black holes observed beyond
redshift of about 6. Extending our previous work, we here study the evolution
of primordial stars growing under such rapid mass accretion until the stellar
mass reaches 10^{4 - 5} Msun. Our stellar evolution calculations show that a
star becomes supermassive while passing through the "supergiant protostar"
stage, whereby the star has a very bloated envelope and a contracting inner
core. The stellar radius increases monotonically with the stellar mass, until
=~ 100 AU for M_* >~ 10^4 Msun, after which the star begins to slowly contract.
Because of the large radius the effective temperature is always less than 10^4
K during rapid accretion. The accreting material is thus almost completely
transparent to the stellar radiation. Only for M_* >~ 10^5 Msun can stellar UV
feedback operate and disturb the mass accretion flow. We also examine the
pulsation stability of accreting SMSs, showing that the pulsation-driven mass
loss does not prevent stellar mass growth. Observational signatures of bloated
SMSs should be detectable with future observational facilities such as the
James Webb Space Telescope. Our results predict that an inner core of the
accreting SMS should suffer from the general relativistic instability soon
after the stellar mass exceeds 10^5 Msun. An extremely massive black hole
should form after the collapse of the inner core.
