Abstract: This paper presents a method for conformal mapping of a two-connected region onto an annulus. The principle of the method is to find a holomorphic function, the real part of which should be a harmonic function satisfying certain boundary conditions.The key for solving the problem is to determine the inner radius of annulus. According to the theory of complex functions we shall determine it from the condition that the line integral predicted along multiple closed paths should be zero.It is then easy to see that the imaginary part can directly be obtained with the aid of Cauchy-Riemann equations. The unknown integral constants can also be derived by using the one-to-one mapping of previous region onto annulus.Without loss of generality, the method may be used to conformally map other two-connected regions onto an annulus.