We report the first observation of the parity-violating gamma-ray asymmetry A_γ^np in neutron-proton capture using polarized cold neutrons incident on a liquid parahydrogen target at the Spallation Neutron Source at Oak Ridge National Laboratory. A_γ^np isolates the ΔI = 1, ³S₁ → ³P₁ component of the weak nucleon-nucleon interaction, which is dominated by pion exchange and can be directly related to a single coupling constant in either the DDH meson exchange model or pionless effective field theory. We measured A_γ^np = [-3.0 ± 1.4(stat) ± 0.2(syst)] × 10^-8, which implies a DDH weak πNN coupling of h_π¹ = [2.6 ± 1.2(stat) ± 0.2(syst)] × 10^-7 and a pionless EFT constant of C^{3S₁→3P₁}/C₀ = [-7.4 ± 3.5(stat) ± 0.5(syst)] × 10^-11 MeV^-1. We describe the experiment, data analysis, systematic uncertainties, and implications of the result.