Sony shares fell sharply as Tokyo's benchmark Nikkei index resumed trade after the Golden Week holiday.
The Japanese consumer electronics company fell almost 4% in early trade.
The company admitted on 3 May, while Japanese markets were closed, that another 25 million users' data had been stolen in a second security breach.
Sony boss Howard Stringer apologised to users "for the inconvenience and concern caused". It was the third apology the firm has made.
Sony had previously said a security breach of its PlayStation Network had lead to 77 million users' data being stolen.
"I know this has been a frustrating time for all of you," Mr Stringer said in a statement on Sony's US PlayStation website.
He also said that there is no confirmed evidence any credit card or personal information had been misused.
Ahead of Mr Stringer's statement some analysts had said security worries could hurt future sales and growth prospects.
"There is a real concern that trust in Sony's business will decline," wrote Kota Ezawa from Citigroup in Japan in a note.
"The network business itself still only makes a small direct contribution to earnings, but we see a potential drop in hardware sales as a concern."Blame wrangle
Sony has blamed an online vigilante group, Anonymous, saying a hacker had gained access to the personal data of more than 100 million online gamers.
Anonymous has denied being involved in the data theft.
"Let's be clear, we are legion, but it wasn't us. You are incompetent Sony," the group said on its website on Thursday.
Sony said on Sunday it had found a file on one of its servers labelled Anonymous and containing the phrase "We are legion", which is used by the group.