We have measured for the first time the cross section of the 2H(n,γ)3H reaction at an energy relevant to big-bang nucleosynthesis by employing a prompt discrete -ray detection method. The outgoing photons have been detected by means of anti-Compton NaI(Tl) spectrometers with a large signal-to-noise ratio. The resulting cross sections are 2.23±0.34,1.99±0.25, and 3.76±0.41µb at En=30.5,54.2, and 531 keV, respectively. At En=30.5 keV the cross section differs from the value reported previously by a factor of 2. Based on the present data the reaction rate has been obtained for temperatures in the range 107-1010 K. The astrophysical impact of the present result is discussed. The obtained cross sections are compared with a theoretical calculation based on the Faddeev approach, which includes meson exchange currents as well as a three-nucleon force