  We introduce the notion of stable representations, -- it is a new class of
the representations of a certain class of groups which defined with positive
definite functions which generalize the classical notion of the characters (or
trace). We give the complete description of this class for infinite symmetric
group ${\frak S}_{\Bbb N}$. It happened the family of the stable
representations of ${\frak S}_{\Bbb N}$ coincides with the set of
representations of the components (left or right) of the admissible
representations of the double-group in the sense of Ol'shanski. In particular
all these are representation of type ${\rm II}$ which was open question for
many years.
