Minegishi, 20, apologized to her fans and swore she did not want to leave the band, despite having violated its production company's guidelines by spending the night with her boyfriend, the BBC reported Friday.
The dalliance became public when she was photographed by a tabloid leaving her beau's flat.
Her nearly 4-minute video apology was posted on AKB48's official website.
"I don't believe just doing this means I can be forgiven for what I did, but the first thing I thought was that I don't want to quit AKB48," the BBC quoted the singer as saying. "If it is possible, I wish from the bottom of my heart to stay in the band. Everything I did is entirely my fault. I am so sorry."