  We perform laser absorption spectroscopy with ultracold $^4$He atoms to
measure the energy intervals between the $2^{3\!} P_2$ level and five levels in
the n = 5 manifold. The laser light perturbs the cold atomic cloud during the
production of Bose-Einstein condensates and decreases the phase space density,
causing a measurable decrease in the number of atoms in the final condensate.
We improve on the precision of previous measurements by at least an order of
magnitude, and report the first observation of the spin-forbidden $2^{3\!}P_2 -
5^{1\!}D_2$ transition in helium. Theoretical transition energies agree with
the observed values within our experimental uncertainty.
