SAN FRANCISCO >> Faced with unknown costs related to two huge data breaches, Yahoo and Verizon Communications announced Tuesday that they had agreed to shave $350 million from the price that Verizon would pay to buy Yahoo’s core internet businesses.

The two companies said they would also share liabilities related to the breaches, which occurred in 2013 and 2014 but were only disclosed last year after the deal was announced.

The revised agreement, now valued at $4.48 billion, paves the way for the deal to proceed to a shareholder vote as early as April, although securities regulators are still assessing how Yahoo disclosed information about the breaches to investors. Yahoo, which is winding down its own investigation of the breaches, will share more details about the incidents and their impact in the next few weeks when it makes required regulatory filings.

Yahoo reiterated on Tuesday that it still expected the deal, originally struck last July, to close by the end of June. The outlines of the new agreement were reported last week.

Consummation of the sale to Verizon would bring closure to a saga that has stretched on for nearly three years as Yahoo’s board and its chief executive, Marissa Mayer, considered various plans to separate the company’s fading internet businesses from its much more valuable investment stakes in Alibaba, China’s largest e-commerce company, and Yahoo Japan, an affiliated company controlled by SoftBank Group.

Yahoo still faces dozens of lawsuits related to the theft of its user data. Hackers stole information, including weakly encrypted passwords, on 1 billion user accounts in 2013, later selling the information online. The following year, another set of hackers, which Yahoo believes were sponsored by a foreign government, stole similar information on more than 500 million accounts.

The two breaches, the largest known hacks of a private corporation, came amid infighting at Yahoo over security investments and safety measures that would inconvenience users.

Verizon and Yahoo, whose internet advertising businesses are tiny compared with juggernauts like Google and Facebook, are eager to move forward on integrating their internet operations.

“We have always believed this acquisition makes strategic sense,” Marni M. Walden, the Verizon executive vice president who oversees product innovation and new businesses, said in a statement. “We look forward to moving ahead expeditiously so that we can quickly welcome Yahoo’s tremendous talent and assets into our expanding portfolio in the digital advertising space.”