We present the concluding result from an Ives-Stilwell-type time dilation experiment using 7Li+ ions confined at a velocity of β=v/c=0.338 in the storage ring ESR at Darmstadt. A Λ-type three-level system within the hyperfine structure of the 7Li+3S1 →3P2 line is driven by two laser beams aligned parallel and antiparallel relative to the ion beam. The lasers' Doppler shifted frequencies required for resonance are measured with an accuracy of <4×10(-9) using optical-optical double resonance spectroscopy. This allows us to verify the special relativity relation between the time dilation factor γ and the velocity β, γ√1-β2=1 to within ±2.3×10(-9) at this velocity. The result, which is singled out by a high boost velocity β, is also interpreted within Lorentz invariance violating test theories.