New astronomical observations point to a nucleosynthesis picture that goes beyond what was accepted until recently. The intermediate "i" process was proposed as a plausible scenario to explain some of the unusual abundance patterns observed in metal-poor stars. The most important nuclear physics properties entering i-process calculations are the neutron-capture cross sections and they are almost exclusively not known experimentally. Here we provide the first experimental constraints on the ^{139}Ba(n,γ)^{140}Ba reaction rate, which is the dominant source of uncertainty for the production of lanthanum, a key indicator of i-process conditions. This is an important step towards identifying the exact astrophysical site of stars carrying the i-process signature.