We present the first results from our ongoing Australia Telescope Compact Array survey of (CO)-C-12(1-0) in Atacama Large Millimeter Array (ALMA)-identified submillimetre galaxies (SMGs) in the Extended Chandra Deep Field South. Strong detections of 12CO(1-0) emission from two SMGs, ALESS 122.1 (z = 2.0232) and ALESS 67.1 (z = 2.1230), were obtained. We estimate gas masses of M-gas similar to 1.3 x 10(11) M-circle dot and M-gas similar to 1.0 x 10(11) M-circle dot for ALESS 122.1 and ALESS 67.1, respectively, adopting alpha(CO) = 1.0. Dynamical mass estimates from the kinematics of the (CO)-C-12(1-0) line yields M-dyn sin(2) i = (2.1 +/- 1.1) x 10(11) M-circle dot and (3.2 +/- 0.9) x 10(11) M-circle dot for ALESS 122.1 and ALESS 67.1, respectively. This is consistent with the total baryonic mass estimates of these two systems. We examine star formation efficiency, using the LFIR versus L '(CO)(1- 0) relation for samples of local ultraluminous infrared galaxies (ULIRGs) and Luminous Infrared Galaxies (LIRGs), and more distant star- forming galaxies, with (CO)-C-12(1- 0) detections. We find some evidence of a shallower slope for ULIRGs and SMGs compared to less luminous systems, but a larger sample is required for definite conclusions. We determine gas-to-dust ratios of 170 +/- 30 and 140 +/- 30 for ALESS 122.1 and ALESS 67.1, respectively, showing that ALESS 122.1 has an unusually large gas reservoir. By combining the 38.1 GHz continuum detection of ALESS 122.1 with 1.4 and 5.5 GHz data, we estimate that the free-free contribution to radio emission at 38.1 GHz is 34 +/- 17 mu Jy, yielding a star formation rate (1400 +/- 700 M-circle dot yr(-1)) consistent with that from the infrared luminosity.