  A formula of Doetsch ({\em Math. Zeitschr.} {\bf 42}, 263 (1937)) is
generalized and used to numerically invert the one-sided Laplace transform
${\hat C}(\beta)$. The necessary input is only the values of ${\hat C}(\beta)$
on the positive real axis. The method is applicable provided that the functions
$\hat{C}(\beta)$ belong to the function space $L^2_\alpha$ defined by the
condition that $ G(x) = e^{x\alpha}\hat{C}(e^x),~ \alpha > 0$ has to be square
integrable. This space includes sums of exponential decays ${\hat
C}(\beta)=\sum_n^{\infty}a_n e^{-\beta E_n}$, e.g. partition functions with
$a_n = 1$. In practice, the inversion algorithm consists of two subsequent fast
Fourier transforms. High accuracy inverted data can be obtained, provided that
the signal is also highly accurate. The method is demonstrated for a harmonic
partition function and resonant transmission through a barrier. We find
accurately inverted functions even in the presence of noise.
