Using methods of differential and integral calculus, we present and discuss the calculation of a fractional Green function associated with the one-dimensional case of the so-called general fractional telegraph equation with one space variable. This is a fractional partial differential equation with constant coefficients. The equation is solved by means of juxtaposition of transforms, i.e., we introduce the Laplace transform in the time variable and the Fourier transform in the space variable. Several particular cases are discussed in terms of the parameters. Some known results are recovered. As a by-product of our main result, we obtain two new relations involving the two-parameter Mittag-Leffler function. (C) 2008 American Institute of Physics.493