  We consider the problem of constructing a Gelfand--Tsetlin basis in
irreducible representations of an infinite-dimensional general linear group.
For a finite-dimensional irreducible representation of a general linear group,
all elements of the Gelfand--Tsetlin basis are parameterized by
Gelfand--Tsetlin schemes. We extend this definition to infinite
Gelfand--Tsetlin schemes, which in turn parameterize elements of the
Gelfand--Tsetlin basis of an irreducible representation of an
infinite-dimensional complete linear group. Using the properties of colimits of
representations with the highest weight, we present an explicit form of the
Gelfand--Tsetlin basis.
