In this paper, the ill-posed problem of attempting to recover the temperature functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A quasi-boundary value regularization method is discussed. An a priori parameter choice rule is discussed and the corresponding error estimate between the exact solution and its regularized approximation is obtained. Moreover, a numerical example is provided to verify our theoretical results.