The origin of the X-ray emission from Herbig Ae/Be stars is not yet known. These intermediate-mass pre-main-sequence stars lie on radiative tracks and are not expected to emit X-rays via solar-like magnetic processes, nor are their winds powerful enough to produce X-rays by radiative wind shocks as in more massive O-type stars. The emission could originate in unseen low-mass companions, or it may be intrinsic to the Herbig stars themselves if they still have primordial magnetic fields or can sustain magnetic activity via a nonsolar dynamo. We present new X-ray observations of the nearby Herbig Ae star HD 104237 (=DX Cha) with XMM-Newton, whose objective is to clarify the origin of the emission. Several X-ray emission lines are clearly visible in the CCD spectra, including the high-temperature Fe Kα complex. The emission can be accurately modeled as a multitemperature thermal plasma with cool (kT < 1 keV) and hot (kT ≳ 3 keV) components. The presence of a hot component is compelling evidence that the X-rays originate in magnetically confined plasma, either in the Herbig star itself or in the corona of an as yet unseen late-type companion. The X-ray temperatures and luminosity (log L_X = 30.5 ergs s^-1) are within the range expected for a T Tauri companion, but high-resolution Chandra and Hubble Space Telescope images constrain the separation of a putative companion to less than 1''. We place these new results into broader context by comparing the X-ray and bolometric luminosities of a sample of nearby Herbig stars with those of T Tauri stars and classical main-sequence Be stars. We also test the predictions of a model that attributes the X-ray emission of Herbig stars to magnetic activity that is sustained by a shear-powered dynamo.