It is well known that the non-linear equations of Malmquist type are more general than the Painleve equations and the Riccati equations in the field of complex differential equations and complex difference equations. In this paper, by using the difference analogues of Nevanlinna theory and the differential field theory, we investigate one type of complex difference equation of Malmquist type with exponential coefficients. We prove that for any transcendental meromorphic solution to the above complex difference equation of Malmquist type, if it is of finite order and has few zeros and poles, its growth should be of the same rate as that of the exponential function ${\rm e}^z$. The obtained result generalizes and complements the complex differential Malmquist Theorem and the complex difference Malmquist Theorem.