ALMA Cycle 2 observations of the long wavelength dust emission in 145 star-forming galaxies are used to probe the evolution of star-forming ISM. We also develop the physical basis and empirical calibration (with 72 low-z and z ~ 2 galaxies) for using the dust continuum as a quantitative probe of interstellar medium (ISM) masses. The galaxies with highest star formation rates (SFRs) at <z> = 2.2 and 4.4 have gas masses up to 100 times that of the Milky Way and gas mass fractions reaching 50 to 80%, i.e. gas masses 1 - 4 times their stellar masses. We find a single high-z star formation law: SFR = 35 M_ mol^0.89 x (1+z)_{z=2}^0.95 x (sSFR)_{MS}^0.23 msun yr^-1 -- an approximately linear dependence on the ISM mass and an increased star formation efficiency per unit gas mass at higher redshift. Galaxies above the Main Sequence (MS) have larger gas masses but are converting their ISM into stars on a timescale only slightly shorter than those on the MS -- thus these starbursts are largely the result of having greatly increased gas masses rather than and increased efficiency for converting gas to stars. At z $> 1$, the entire population of star-forming galaxies has $sim$ 2 - 5 times shorter gas depletion times than low-z galaxies. These shorter depletion times indicate a different mode of star formation in the early universe -- most likely dynamically driven by compressive, high-dispersion gas motions -- a natural consequence of the high gas accretion rates.
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