We consider one-dimensional, weakly nonlinear first and second sound waves in He II. The first correction to the shock speed is computed for each sound mode. The expressions obtained are exact with respect to the coefficient of thermal expansion beta. It is shown that the commonly made assumption of negligible beta can lead to significant error in the shock speeds for first sound away from the lambda-line. This contrasts with the calculation of the linear sound speeds and the shock speed for second sound where the beta=0 approximation yields accurate results. Near the lambda-line, the exact expressions for both modes are seen to contain fundamentally different singularities than those found in the commonly employed beta=0 theory. (C) 1996 American Institute of Physics.