The Magellanic Bridge is a tidal feature located between both Magellanic Clouds, containing young stars formed in situ. Its proximity allows high-resolution studies of molecular gas, dust and star formation in a tidal, low metallicity environment. Our goal is to characterize gas and dust emission in Magellanic Bridge A, the source with the highest 870um excess of emission found in single dish surveys. Using the ALMA telescope including the Morita Array, we mapped with sub-parsec resolution a 3arcmin, field of view centered on the Magellanic Bridge A molecular cloud, in 1.3mm continuum emission and ^12^CO(2-1 line emission. This region was also mapped in continuum at 870um and in ^12^CO(2-1) line emission at ~6pc resolution with the APEX telescope. To study its dust properties, we also use archival Herschel and Spitzer data. We combine the ALMA and APEX ^12^CO(2-1) line cubes to study the molecular gas emission. Magallanic Bridge A breaks up into two distinct molecular clouds in dust and ^12^CO(2-1) emission, which we call North and South. Dust emission in the North source, according to our best parameters from fitting the far-infrared fluxes, is ~3K colder than in the South source in correspondence to its less developed star formation. Both dust sources present large submillimeter excesses in LABOCA data: according to our best fits the excess over the modified blackbody (MBB) fit to the Spitzer/Herschel continuum is E(870um)~7 and E(870um)~3 for the North and South sources respectively. Nonetheless, we do not detect the corresponding 1.3mm continuum with ALMA. Our limits are compatible with the extrapolation of the MBB fits and therefore we cannot independently confirm the excess at this longer wavelength. The ^12^CO(2-1) emission is concentrated in two parsec-sized clouds with virial masses around 400 and 700M_{sun} each. Their bulk volume densities are n(H_2)~0.7-2.6x10^3^cm^-3^, larger than typical bulk densities of Galactic molecular clouds. The ^12^CO luminosity to H_2_ mass conversion factor {alpha}CO is 6.5 and 15.3M_{sun}/(K.(km/s)pc^2^) for the North and South clouds, calculated using their respective virial masses and ^12^CO(2-1) luminosities. Gas mass estimates from our MBB fits to dust emission yields masses M~1.3x10^3^M{sun} and 2.9x10^3^M{sun}_ for North and South respectively, a factor of ~4 larger than the virial masses we infer from ^12^CO.
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