  The first observation of the decays $\Lambda_b^0 \to \chi_{c1} p K^-$ and
$\Lambda_b^0 \to \chi_{c2} p K^-$ is reported using a data sample corresponding
to an integrated luminosity of $3.0$ fb$^{-1}$, collected by the LHCb
experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV. The
following ratios of branching fractions are measured
  \begin{eqnarray*}
  \frac{{\cal B}(\Lambda_b^0 \to \chi_{c1} p K^-)}{{\cal B}(\Lambda_b^0 \to
J/\psi p K^-)} = 0.242 \pm 0.014 \pm 0.013 \pm 0.009,
  \frac{{\cal B}(\Lambda_b^0 \to \chi_{c2} p K^-)}{{\cal B}(\Lambda_b^0 \to
J/\psi p K^-)} = 0.248 \pm 0.020 \pm 0.014 \pm 0.009,
  \frac{{\cal B}(\Lambda_b^0 \to \chi_{c2} p K^-)}{{\cal B}(\Lambda_b^0 \to
\chi_{c1} p K^-)} = 1.02 \pm 0.10 \pm 0.02 \pm 0.05,
  \end{eqnarray*} where the first uncertainty is statistical, the second
systematic and the third due to the uncertainty on the branching fractions of
the $\chi_{c1}\to J/\psi\gamma$ and $\chi_{c2} \to J/\psi\gamma$ decays. Using
both decay modes, the mass of the $\Lambda_b^0$ baryon is also measured to be
$m_{\Lambda_b^0} = 5619.44 \pm 0.28 \pm 0.26$ MeV/$c^2$, where the first and
second uncertainties are statistical and systematic, respectively.
