Let $f$ be a transcendental meromorphic function of finite order in the plane such that $f^{(m)}$ has finitely many zeros for some positive integer $mgeq 2.$ Suppose that $f^{(k)}$ and $f$ share $a$ CM, where $kgeq 1$ is a positive integer, $a
eq 0$ is a finite complex value. Then $f$ is an entire function such that $f^{(k)}-a=c(f-a),$ where $c
eq 0$ is a nonzero constant. The results in this paper are concerning a conjecture of Br"{u}ck [5]. An example is provided to show that the results in this paper, in a sense, are the best possible.

Keywords: Meromorphic functions, Order of growth, Shared values, Uniqueness theorems