  The $1\mathrm{s}n\mathrm{p}\,^3\mathrm{P}_J$ fine structure of high Rydberg
states in helium has been measured by microwave spectroscopy of single-photon
transitions from $1\mathrm{s}n\mathrm{s}\,^3\mathrm{S}_1$ levels in pulsed
supersonic beams. For states with principal quantum numbers in the range from
$n=34$ to 36, the $J = 0 \rightarrow 2$ and $J = 1 \rightarrow 2$ fine
structure intervals were both observed. For values of $n$ between 45 and 51
only the larger $J = 0 \rightarrow 2$ interval was resolved. The experimental
results are in good agreement with theoretical predictions. Detailed
characterization of residual uncancelled electric and magnetic fields in the
experimental apparatus, and calculations of the Stark and Zeeman structures of
the Rydberg states in weak fields, were used to quantify systematic
contributions to the uncertainties in the measurements.
