"Once calm is fully restored and the violence ends, a broader cease-fire will have to address all the underlying causes of conflict, including the full opening of crossings, Palestinian reconciliation and an end to weapons smuggling."
How Middle East has changed since last Gaza conflict
Regev said the deal calls for talks to begin Thursday on easing economic restrictions on Gaza.
"If the border is quiet, that enables us to be more forthcoming," he said.
Clinton, who shuttled among Israel, the West Bank and Egypt to help negotiate the deal, said the United States will continue to work with regional partners to implement and expand the agreement.
Obama spoke Wednesday morning with Morsy, thanking the Egyptian leader for his leadership in negotiating the proposal.
As recently as Wednesday afternoon, Hamas officials had been calling for more strikes against Israel, while that country's military continued to press its campaign against what it said were suspected rocket-launching sites and "terrorist hideouts."
The cease-fire talks, held in the West Bank, Israel and Cairo, continued despite a lunchtime bus attack near the Defense Ministry headquarters in Tel Aviv. At least two bombs were planted on the bus, Israeli police spokesman Micky Rosenfeld said. One of the bombs exploded, blowing out the windows of the bus, he said.
Rosenfeld said 24 people were wounded, three of them seriously. Aviva Shemer with Ichilov Hospital said pedestrians were among the injured.
Police said they were seeking at least one and possibly two suspects.
Hamas put its own spin on the attack in a banner on al-Aqsa, calling it "a natural response to the massacre of the al-Dalou family and targeting of innocent Palestinian civilians."
Nine members of the al-Dalou family died Sunday in an Israeli airstrike, provoking outrage among Palestinians.
"We told you #IDF that our blessed hands will reach your leaders and soldiers wherever they are," the al Qassam Brigades, the military wing of Hamas, said on Twitter. "You opened the Gates of Hell on Yourselves."
Diplomats said they were hoping to avoid a repeat of 2008 and 2009, when at least 1,400 people died as Israeli troops invaded Gaza after similar rocket attacks.