We present a broadband imaging and spectral study of the radio-bright supernova remnant (SNR) 3C 397 with ROSAT , ASCA, and RXT E. A bright X-ray spot seen in the HRI image hints at the presence of a pulsar-powered component and gives this SNR a composite X-ray morphology. Combined ROSAT and ASCA imaging shows that the remnant is highly asymmetric, with its X-ray emission peaking at the western lobe. The hard-band images obtained with the ASCA Gas Imaging Spectrometer show that much of the hard X-ray emission arises from the western lobe, associated with the SNR shell, with little hard X-ray emission associated with the central hot spot. The spectrum from 3C 397 is heavily absorbed and dominated by thermal emission with emission lines evident from Mg, Si, S, Ar and Fe. Single-component models fail to describe the X-ray spectrum, and at least two components are required : a soft component characterized by a low temperature and a large ionization timescale, and a hard component required to account for the Fe-K emission line and characterized by a much lower ionization timescale. We use a set of nonequilibrium ionization (NEI) models (Borkowski et al., in preparation), and Ðnd that the Ðtted parameters are robust. The temperatures from the soft and hard components are D0.2 keV and D1.6 keV respectively. The corresponding ionization timescales being the pren 0 t (n 0 shock hydrogen density) are D6 ] 1012 cm~3 s and D6 ] 1010 cm~3 s, respectively. The large of n 0 t the soft component suggests it is approaching ionization equilibrium ; thus it can be Ðt equally well with a collisional equilibrium ionization model. The spectrum obtained with the Proportional Counter Array (PCA) of RXT E is contaminated by emission from the Galactic ridge, with only D15% of the count rate originating from 3C 397 in the 5È15 keV range. The PCA spectrum allowed us to conÐrm the thermal nature of the hard X-ray emission. A third component originating from a pulsar-driven component is possible, but the contamination of the source signal by the Galactic ridge did not allow us to determine its parameters or Ðnd pulsations from any hidden pulsar. We discuss the X-ray spectrum in the light of two scenarios : a young ejecta-dominated remnant of a core-collapse SN, and a middle-aged SNR expanding in a dense ISM. In the Ðrst scenario, the hot component arises from the SNR shell, and the soft component from an ejecta-dominated component. 3C 397 would be a young SNR (a few thousand years old), but intermediate in dynamical age between the young historical shells (like Tycho or Kepler), and those that are well into the Sedov phase of evolution (like Vela). In the second scenario, the soft component represents the blast wave propagating in a dense medium, and the hard component is associated with hot gas encountering a fast shock, or arising from thermal conduction. In this latter scenario, the SNR would be Dtwice as old, and transitioning into the radiative phase. The current picture we present in this paper is marginally consistent with thi...