  Direct collapse of supermassive stars (SMSs) is a possible pathway for
generating supermassive black holes in the early universe. It is expected that
an SMS could form via very rapid mass accretion with Mdot ~ 0.1 - 1 Msun/yr
during the gravitational collapse of an atomic-cooling primordial gas cloud. In
this paper we study how stars would evolve under such extreme rapid mass
accretion, focusing on the early evolution until the stellar mass reaches 1000
Msun. To this end we numerically calculate the detailed interior structure of
accreting stars with primordial element abundances. Our results show that for
accretion rates higher than 0.01 Msun/yr, stellar evolution is qualitatively
different from that expected at lower rates. While accreting at these high
rates the star always has a radius exceeding 100 Rsun, which increases
monotonically with the stellar mass. The mass-radius relation for stellar
masses exceeding ~ 100 Msun follows the same track with R_* \propto M_*^0.5 in
all cases with accretion rates > 0.01 Msun/yr; at a stellar mass of 1000 Msun
the radius is about 7000 Rsun (~= 30 AU). With higher accretion rates the onset
of hydrogen burning is shifted towards higher stellar masses. In particular,
for accretion rates exceeding Mdot > 0.1 Msun/yr, there is no significant
hydrogen burning even after 1000 Msun have accreted onto the protostar. Such
"supergiant" protostars have effective temperatures as low as Teff ~= 5000 K
throughout their evolution and because they hardly emit ionizing photons, they
do not create an HII region or significantly heat their immediate surroundings.
Thus, radiative feedback is unable to hinder the growth of rapidly accreting
stars to masses in excess of 1000 Msun, as long as material is accreted at
rates Mdot > 0.01 Msun/yr.
