With properties of non-negative operator and Lax-Milgram theorem, we transform an ill-posed 1D heat conduction problem into a well-posed second kind Volterra equation, and introduce a stable and fast algorithm. Related theoretical analysis is shown. For the determination of regularization parameter, the algorithm employs two posterior strategies, which can be realized quickly whether the error level of input data are known or unknown. Numerical tests demonstrate merit of the algorithm.