We present the first lattice determination of the two lowest Gegenbauer moments of the leading-twist pion and kaon light-cone distribution amplitudes with full control of all errors: a(2)(pi) = 0.101(-24)(+24) for the pion; a(1)(K) = 0.0533(-34)(+34) and a(2)(K) = 0.090(-19)(+19) for the kaon. The calculation is carried out on 35 different CLS ensembles with N-f = 2 + 1 flavors of dynamical Wilson-clover fermions. These cover a multitude of pion and kaon mass combinations (including the physical point) and 5 different lattice spacings down to a = 0.039 fm. The momentum smearing technique and a new operator basis are employed to reduce statistical fluctuations and to improve the overlap with the ground states. The results are obtained from a combined chiral and continuum limit extrapolation that includes three separate trajectories in the quark mass plane.