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Carbon-bearing molecules, particularly CO, have been widely used as tracers of molecular gas in the interstellar medium (ISM). In this work, we aim to study the properties of molecules in diffuse, cold environments, where CO tends to be underabundant and/or subthermally excited. We performed one of the most sensitive (down to τCOrms ∼ 0.002 and τHCO+rms ∼ 0.0008) submillimeter molecular absorption line observations toward 13 continuum sources with the ALMA. CO absorption was detected in diffuse ISM down to Av < 0.32 mag and HCO+ was detected down to Av < 0.2 mag , where atomic gas and dark molecular gas start to dominate. Multiple transitions measured in absorption toward 3C454.3 allow for a direct determination of excitation temperatures Tex of 4.1 and 2.7 K, for CO and for HCO+, respectively, which are close to the cosmic microwave background and explain their being undercounted in emission surveys. A stronger linear correlation was found between NHCO+ and NH2 (Pearson correlation coefficient P ∼ 0.93) than that of NCO and NH2 (P ∼ 0.33), suggesting HCO+ is a better tracer of H2 than CO in diffuse gas. The derived CO-to-H2 conversion factor (the CO X-factor) of (14 ± 3) × 1020 cm-2 (K km s-1) -1 is approximately six times larger than the average value found in the Milky Way.