Rotational spectra have been recorded for six isotopomers of the trimethylamine-hydrogen sulfide complex using a Fourier-transform microwave spectrometer. The spectra are characteristic of a symmetric top (B+C)/2=1395.363 (1) MHz, and are indicative of free internal rotation of H2S about the trimethylamine symmetry axis. A structure with a single, linear hydrogen bond, with an S-N distance of 3.36 (5) Angstrom, best reproduces the moments of inertia of six isotopomers, including three distinct deuterated complexes. The experimental structure is compared to an ab initio structure optimized at the MP2/6-31+G(d,p) level which predicts an S-N distance of 3.328 Angstrom. MP2/aug’-cc-pVTZ calculations were used to determine the binding energy of the complex (Delta E-e=-5.8 kcal/mol) and the barrier to an internal tunnelling motion which exchanges the two H2S hydrogens (3.0 kcal/mol).