Recently, exceptional points, a degeneracy of open wave systems, have been observed in photonics, acoustics, and electronics. They have mainly been realized as a degeneracy of resonances; however, a degeneracy associated with the absorption of waves can exhibit distinct and interesting physical features. Here, we demonstrate such an absorbing exceptional point by engineering degeneracies in the absorption spectrum of dissipative optical microcavities. We experimentally distinguished the conditions to realize an absorbing exceptional point versus a resonant exceptional point. Furthermore, when the optical loss was tuned to achieve perfect absorption at an absorbing exceptional point, we observed its signature, an anomalously broadened line shape in the absorption spectrum. The distinct scattering properties of the absorbing exceptional point create opportunities for both fundamental study and applications of non-Hermitian degeneracies.