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HD 135344 AB is a young visual binary system that is best known for the protoplanetary disk around the secondary star. The circumstellar environment of the A0-type primary star, on the other hand, is already depleted. HD 135344 A is therefore an ideal target for the exploration of recently formed giant planets because it is not obscured by dust.<\/jats:p>\n<jats:p><jats:italic>Aims<\/jats:italic>. We searched for and characterized substellar companions to HD 135344 A down to separations of about 10 au.<\/jats:p>\n<jats:p><jats:italic>Methods<\/jats:italic>. We observed HD 135344 A with VLT\/SPHERE in the <jats:italic>H<\/jats:italic>23 and <jats:italic>K<\/jats:italic>12 bands and obtained <jats:italic>YJ<\/jats:italic> and <jats:italic>YJH<\/jats:italic> spectroscopy. In addition, we carried out VLTI\/GRAVITY observations for the further astrometric and spectroscopic confirmation of a detected companion.<\/jats:p>\n<jats:p><jats:italic>Results<\/jats:italic>. We discovered a close-in young giant planet, HD 135344 Ab, with a mass of about 10 <jats:italic>M<\/jats:italic><jats:sub>J<\/jats:sub>. The multi-epoch astrometry confirms the bound nature based on common parallax and common proper motion. This firmly rules out the scenario of a non-stationary background star. The semi-major axis of the planetary orbit is approximately 15-20 au, and the photometry is consistent with that of a mid L-type object. The inferred atmospheric and bulk parameters further confirm the young and planetary nature of the companion.<\/jats:p>\n<jats:p><jats:italic>Conclusions<\/jats:italic>. HD 135344 Ab is one of the youngest directly imaged planets that has fully formed and orbits on Solar System scales. 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