  The external pollution of the first stars in the Galaxy is investigated. The
first stars were born in clouds composed of the pristine gas without heavy
elements. These stars accreted gas polluted with heavy elements while they
still remained in the cloud. As a result, it is found that they exhibit a
distribution with respect to the surface metallicity. We have derived the
actual form of this distribution function. This metallicity distribution
function strongly suggests that the recently discovered most metal-deficient
star HE0107-5240 with [Fe/H]=-5.3 was born as a metal-free star and accreted
gas polluted with heavy elements. Thus the heavy elements such as Fe in
HE0107-5240 must have been supplied from supernovae of later generations
exploding inside the cloud in which the star had been formed. The elemental
abundance pattern on the surface of stars suffering from such an external
pollution should not be diverse but exhibit the average pattern of numerous
supernovae. Future observations for a number of metal-deficient stars with
[Fe/H]<-5 will be able to prove or disprove this external pollution scenario.
Other possibilities to produce a star with this metallicity are also discussed.
