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The correct prediction of the abundances of the light nuclides produced during the epoch of Big Bang Nucleosynthesis (BBN) is one of the main topics of modern cosmology. For many of the nuclear reactions that are relevant for this epoch, direct experimental cross section data are available, ushering the so-called “age of precision”. The present work addresses an exception to this current status: the 2H(α,γ)6Li reaction that controls 6Li production in the Big Bang. Recent controversial observations of 6Li in metal-poor stars have heightened the interest in understanding primordial 6Li production. If confirmed, these observations would lead to a second cosmological lithium problem, in addition to the well-known 7Li problem. In the present work, the direct experimental cross section data on 2H(α,γ)6Li in the BBN energy range are reported. The measurement has been performed deep underground at the LUNA (Laboratory for Underground Nuclear Astrophysics) 400 kV accelerator in the Laboratori Nazionali del Gran Sasso, Italy. The cross section has been directly measured at the energies of interest for Big Bang Nucleosynthesis for the first time, at E c m = 80, 93, 120, and 133 keV. Based on the new data, the 2H(α,γ)6Li thermonuclear reaction rate has been derived. Our rate is even lower than previously reported, thus increasing the discrepancy between predicted Big Bang 6Li abundance and the amount of primordial 6Li inferred from observations.