A generalization of Doetsch's formula [Math. Z. 42, 263 (1937)] is derived to develop a stable numerical inversion of the one-sided Laplace transform (C) over cap (beta). The necessary input is only the values of C ( b) on the positive real axis. The method is applicable provided that the functions (C) over cap (beta) belong to the function space L-alpha(2) defined by the condition that G(x) = e(x alpha)(C) over cap(e(x)), alpha>0 has to be square integrable. The inversion algorithm consists of two sequential Fourier transforms where the second Fourier integration requires a cutoff, whose magnitude depends on the accuracy of the data. For high accuracy data, the cutoff tends to infinity and the inversion is very accurate. The presence of noise in the signal causes a lowering of the cutoff and a lowering of the accuracy of the inverted data. The optimal cutoff value is shown to be one which leads to an inversion which remains consistent with the original data and its noise level. The method is demonstrated for some model problems: a harmonic partition function, resonant transmission through a barrier, noisy correlation functions, and noisy Monte Carlo generated data for tunneling coefficients obtained via the recently introduced quantum transition state theory (QTST).