To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index. It is required to define a function Φ(z), analytic and bounded in the complex plane z, cut down on positive real semi-axis L+ , if the edge condition Φ+(t) = G(t)Φ−(t), t ∈ L+ is fulfilled, where Φ+(t),     Φ− (t) are limit values of the function Φ(z), as z → tcorrespondingly on the left and on the right,G(t) is a given function, for which argument argG(t) = ν− tρ +ν(t), t ∈ L+  holds, here ν− , ρ are given numbers, ν− > 0, 1/2< ρ < 1, and ln|G(t)|, ν(t) are functions which satisfy the Holder condition. It is admitted that G(t) = 1 at t ∈ (−∞,0). The functions E+ (z) = e(α+iβ)zρ , 0 ≤ argz ≤ π, E− (z) = e(α−iβ)zρ , −π ≤ argz ≤ 0 are used to avoid infinite gap of the argG(t), by the selection of real numbers α, β.