The talks offered a "possibility for an island of stability," in the region, Biden said in remarks following the closed-door discussions with Peres. "[And] that [possibility] is between the Palestinian people and the Israeli people in two secure states respecting one another's sovereignty and security."
Biden said he and President Obama "believe that this is one of those opportunities, one of those moments in history where it has to be seized."
Peres said Israel would "continue full steam ahead" with the peace talks "with the profound belief that we can overcome remaining gaps."
Palestinians, Peres said, must understand that Israel is "not their enemy," and that terror is "destroying their fabric."
Peres called Palestinian President Mahmoud Abbas "a serious partner," and also "the right one."
He also thanked Biden and Obama for supporting U.S. Secretary of State John Kerry and "energizing" him on his mission.
Biden and Netanyahu had a working dinner at the prime minister's residence. Reporters were told that the dinner would be a one-on-one event after an initial group meeting, the White House media poll said.
Peres noted the United States came to the Middle East because it didn't have enough oil.
Now, Peres said, "you have enough oil. You come for peace; you do not have enough peace."
"Peace negotiations are being reopened [and] problems are being reopened," Peres said, adding, "every day is a meaningful day. More and more both sides are taking it seriously."
Notable deaths of 2014 [PHOTOS]