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 .