Experiments conducted on the J-TEXT tokamak have provided the first evidence that the Beta-induced Alfvén Eigenmode (BAE) is localized inside the isolated helical flux tube of its edge m/n = 3/1 magnetic island. The observations show that the BAE forms a standing wave inside the magnetic island, with its nodes located at the X- and O-points of the magnetic island. When the island is cut open by contact with the limiter plates, the BAE is found to remain inside the remnant closed island in the Scrape-Off Layer, but its amplitude decreases as the width of the remnant island becomes smaller.