We report on a recent similar to 150 ks long Chandra observation of the ultraluminous infrared galaxy merger NGC 6240, which allows a detailed investigation of the diffuse galactic halo. Extended soft X-ray emission is detected at the 3 sigma confidence level over a diamond-shaped region with projected physical size of similar to 110 x 80 kpc, and a single-component thermal model provides a reasonably good fit to the observed X-ray spectrum. The hot gas has a temperature of similar to 7.5 million K, an estimated density of 2.5 x 10(-3) cm(-3), and a total mass of similar to 10(10)M(circle dot), resulting in an intrinsic 0.4-2.5 keV luminosity of 4 x 10(41) erg s(-1). The average temperature of 0.65 keV is quite high to be obviously related to either the binding energy in the dark-matter gravitational potential of the system or the energy dissipation and shocks following the galactic collision, yet the spatially resolved spectral analysis reveals limited variations across the halo. The relative abundance of the main alpha-elements with respect to iron is several times the solar value, and nearly constant as well, implying a uniform enrichment by type II supernovae out to the largest scales. Taken as a whole, the observational evidence is not compatible with a superwind originated by a recent, nuclear starburst, but rather hints at widespread, enhanced star formation proceeding at a steady rate over the entire dynamical timescale (similar to 200 Myr). The preferred scenario is that of a starburst-processed gas component gently expanding into, and mixing with, a pre-existing halo medium of lower metallicity (Z similar to 0.1 solar) and temperature (kT similar to 0.25 keV). This picture cannot be probed more extensively with the present data, and the ultimate fate of the diffuse, hot gas remains uncertain. Under some favorable conditions, at least a fraction of it might be retained after the merger completion, and evolve into the hot halo of a young elliptical galaxy.