We applied the Maximum Likelihood method, as an image reconstruction algorithm, to the BAT X-ray Survey (BXS). This method was specifically designed to preserve the full statistical information in the data and to avoid mosaicking of many exposures with different pointing directions, thus reducing systematic errors when co-adding images. We reconstructed, in the 14-170 keV energy band, the image of a 90x90 deg$^2$ sky region, centered on (RA,DEC)=105$^{\circ}$,-25$^{\circ}$, which BAT surveyed with an exposure time of $\sim1$ Ms (in Nov. 2005). The best sensitivity in our image is $\sim0.85$ mCrab or $2.0\times 10^{-11}$ erg cm$^{-2}$. We detect 49 hard X-ray sources above the 4.5 $��$ level; of these, only 12 were previously known as hard X-ray sources ($>$15 keV). Swift/XRT observations allowed us to firmly identify the counterparts for 15 objects, while 2 objects have Einstein IPC counterparts \citep{harris90}; in addition to those, we found a likely counterpart for 13 objects by correlating our sample with the ROSAT All-Sky Survey Bright Source Catalog \citep{voges99}. 7 objects remain unidentified. Analysis of the noise properties of our image shows that $\sim75$% of the area is surveyed to a flux limit of $\sim$1 mCrab. This study shows that the coupling of the Maximum Likelihood method to the most sensitive, all-sky surveying, hard X-ray instrument, BAT, is able to probe for the first time the hard X-ray sky to the mCrab flux level. The successful application of this method to BAT demonstrates that it could also be applied with advantage to similar instruments like INTEGRAL-IBIS.

35 pages, 10 figures, Accepted for publication in ApJ. 1st paper in a series of three