The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let f be a nonconstant meromorphic function and L a nonconstant linear differential polynomial generated by f. Suppose that a = a(z) (6≡ 0,∞) is a small function of f. If f − a and L− a share 0 CM and (k + 1)N(r,∞; f) + N(r, 0; f ′ ) + Nk(r, 0; f ′ ) < λT (r, f′ ) + S(r, f′ ) for some real constant λ ∈ (0, 1), then f − a = (1 + c/a)(L − a), where c is a constant and 1 + c/a 6≡ 0.