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Ultrathin Co/Ru/Co trilayers are investigated experimentally by magnetization curves and magnetic-force microscopy (MFM). Emphasis is on the domain-wall fine structure of antiphase domain walls in the films. The trilayers are produced by sputtering and consist of two Co layers of equal thickness (5 nm), exchange-coupled through a Ru layer of variable thickness. The sign and magnitude of the interlayer exchange are tuned by the thickness of the Ru interlayer. The exchange and its distribution are investigated by measurements of the static magnetization curves. For a Ru thickness of 0.4 nm, the exchange is predominantly antiferromagnetic and the MFM images show fairly immobile domain walls. Micromagnetic model calculations yield immobile antiphase domain walls whose thickness decreases with increasing magnetic field but is typically of the order of 100 nm in agreement with experiment.