We present near-infrared (near-IR) spectroscopy of three extremely red objects (EROs) using the OHS/CISCO spectrograph at the Subaru Telescope. One target exhibits a strong emission line, which we identify as Ha at z = 1.34. Using new and existing ground-based optical and near-IR imaging, and archival Hubble Space Telescope imaging, we argue that this target is essentially an elliptical galaxy, with an old stellar population of around 4 x 10(11) M-circle dot, but having a dust-enshrouded star-forming component with a star formation rate (SFR) of some 50-100 M-circle dot yr(-1). There is no evidence that the galaxy contains an active galactic nucleus. Analysis of a further two targets, which do not exhibit any features in our near-IR spectra, suggests that one is a quiescent galaxy in the redshift range 1.2 < z < 1.6, but that the other cannot be conclusively categorized as either star-forming or quiescent. Even though our first target has many of the properties of an old elliptical, the ongoing star formation means that it cannot have formed all of its stellar population at high redshift. While we cannot infer any robust values for the SFR in ellipticals at z > 1 from this one object, we argue that the presence of an object with such a high SFR in such a small sample suggests that a non-negligible fraction of the elliptical galaxy population may have formed a component of their stellar population at redshifts z similar to 1-2. We suggest that this is evidence for ongoing star formation in the history of elliptical galaxies.