  The homogeneous steady state of a fluid of inelastic hard spheres immersed in
a bath of elastic hard spheres kept at equilibrium is analyzed by means of the
first Sonine approximation to the (spatially homogeneous) Enskog--Boltzmann
equation. The temperature of the granular fluid relative to the bath
temperature and the kurtosis of the granular distribution function are obtained
as functions of the coefficient of restitution, the mass ratio, and a
dimensionless parameter $\beta$ measuring the cooling rate relative to the
friction constant. Comparison with recent results obtained from an iterative
numerical solution of the Enskog--Boltzmann equation [Biben et al., Physica A
310, 308 (202)] shows an excellent agreement. Several limiting cases are also
considered. In particular, when the granular particles are much heavier than
the bath particles (but have a comparable size and number density), it is shown
that the bath acts as a white noise external driving. In the general case, the
Sonine approximation predicts the lack of a steady state if the control
parameter $\beta$ is larger than a certain critical value $\beta_c$ that
depends on the coefficient of restitution and the mass ratio. However, this
phenomenon appears outside the expected domain of applicability of the
approximation.
