At thirteen he went as a choir-boy of the Benedictine Abbey of Saints Ulrich and Afra in Augsburg and was a pupil for six years in the Jesuit gymnasium of the same city. He entered the Abbey of Saints Ulrich and Afra as a novice, May 13, 1775, and was ordained a priest, September 18, 1779.
In 1785 he was made head librarian of the abbey. He arranged and catalogued the library and made known to scholars the rarities it contained through the fine descriptions he gave of its early printed books and manuscripts in two works which he published while librarian. These publications were: Notitia historico-litteraria de libris ab artis typographiae inventione usque ad annum 1479 impressis, in bibliotheca monasterii ad SS. Udalricum et Afram Augustae extantibus. Pars I: Augs. Vindel. 1788. Pars II: Notitia . . . libros complectens ab anno usque ad annum 1500 inclusive impressos. Ibidem, 1789 and Notitia Historico-litteraria de codicibus manuscriptis in blbliotheca liberi ac imperialis monasterii O. S. Benedicti ad SS. Udalricum et Afram extantibus. Aug. Vindel., 6 partes 1791-1796.
After the abbey was dissolved as a result of secularization, and its building converted into a barrack in 1806, Braun lived with a number of fellow-members of the order in the house of Kaufmanns Fusami near the church of St. Ulrich. In these new surroundings he endeavoured to observe the rules of the order as far as possible, gave assistance in pastoral work, and devoted himself to the study of the history of the Diocese of Augsburg and its suppressed monastic foundations. He was made a foreign member of the Bavarian Academy of Sciences, August 3, 1808, which honour he accepted, but he declined to settle in Munich.
Among his historical writings the following may be mentioned: Geschichte der Bischöfe von Augsburg, chronologisch und diplomatisch verfasst (4 vols, Augsburg, 1813-15); Codex diplomaticus monasterii S. Udalrici et Afrae notis illustratus issued as volumes XXII and XXIII of the Monumenta Boica, (Munich, 1814-15); Geschichte der Kirche und des Stiftes der hll. Ulrich und Afra in Augsburg (Augsburg, 1817); Historisch-topographische Beschreibung, der Diocese Augsburg, 2 vols. (Augsburg, 1823); Die Domkirche zu Augsburg und der hohere und niedere Klerus an derselben (Augsburg, 1829). Braun bequeathed his manuscripts, which were concerned chiefly with the history of the religious foundations and monastic houses of the Diocese of Augsburg, to the diocesan archives.