Washington (CNN) -- Direct talks between Israeli and Palestinian leaders kicked off again Thursday with pledges of support from leaders on both sides -- as well as the United States -- for a peace process leading to a comprehensive settlement within one year.

Israeli Prime Minister Benjamin Netanyahu and Palestinian leader Mahmoud Abbas also quickly agreed to meet again on September 14 and 15 in the Middle East and then roughly every two weeks thereafter, according to former Sen. George Mitchell, President Barack Obama's special envoy for Middle East peace.

Both Abbas and Netanyahu condemned violence targeting innocent civilians in their region and pledged to work together to maintain security, Mitchell said.

They also reiterated their mutual support for the goal of "two states for two peoples" as part of a resolution of "all core issues" at the heart of the Israeli-Palestinian conflict, he added.

They agreed to work toward the completion of a framework agreement designed to bolster an "atmosphere of trust" and pave the way toward a comprehensive treaty and a "lasting peace," the special envoy said.

Mitchell's announcement came shortly after Netanyahu and Abbas sat down with Secretary of State Hillary Clinton at the State Department.

Clinton, who sat at the middle of a U-shaped table flanked by Netanyahu, Abbas and other senior negotiators, told reporters that the meeting was part of a move "towards a future of peace and dignity that only (Israelis and Palestinians) can create."

The United States "cannot and will not impose a solution," she declared, though she asserted that a two-state solution is the only viable resolution to the conflict.

The U.S. government will be an "active" partner in the renewed peace process, Clinton promised. It's in America's national security interest to find a solution to the conflict, she said.

Clinton, whose husband tried but failed to reach a comprehensive deal in the final year of his administration, said that "we've been here before and know how difficult the road ahead" can be. Opponents of a deal will try "to sabotage this process," she warned.

For his part, Netanyahu said that he sees in Abbas "a partner for peace."

The Israeli leader said that reaching a lasting peace will require "mutual and painful concessions from both sides." But Israel is prepared to go "a long way in a short time" to reach a deal, he said.

Abbas voiced his belief that the current negotiations "should, within a year, lead to an arrangement" for peace. Palestinian negotiators are prepared to "work on all the final status issues," he said.

Abbas used the occasion to, among other things, call on Israel to lift its blockade of Gaza.

Thursday's talks followed a working dinner Wednesday night at the White House with Obama, Abbas, Netanyahu, Egyptian President Hosni Mubarak and Jordan's King Abdullah II.

"I am hopeful -- cautiously hopeful, but hopeful -- that we can achieve the goal that all four of these leaders articulated," Obama said Wednesday.

Clinton and Middle East Quartet Representative Tony Blair also attended the dinner. The Quartet consists of the United States, Russia, the United Nations and the European Union.

While all of the major players have struck an optimistic tone, several key hurdles are confronting negotiators.

One immediate threat to the talks is the September 26 expiration of Israel's 10-month freeze on settlement construction in the West Bank. Palestinian leaders have indicated they may walk out of the talks if the freeze is not extended. Some members of Netanyahu's coalition government, on the other hand, have said they will leave the coalition if the freeze continues.

Another roadblock is the Palestinian view that any two-state solution must include a handover of all the land Israel captured in the 1967 war, along with East Jerusalem as the Palestinian capital.

While Netanyahu has expressed openness about a Palestinian state, he also has voiced strong opposition to a Palestinian takeover of East Jerusalem.

Hamas control of Gaza also remains a "major problem," Chief Palestinian Negotiator Saeb Erakat said earlier this week.

Leaders of Hamas are frequently in conflict with the more moderate Abbas and his Fatah organization, which has the upper hand in the West Bank. Hamas spokesman Sami Abu Zuhri, speaking to reporters in Gaza Thursday, said Abbas has no political legitimacy and no right to represent the Palestinian people.

While Gaza is generally considered to be part of any future Palestinian state, Hamas has refused to recognize Israel's right to exist and is not a part of the talks. Zuhri said that "resistance operations" will continue against Israelis despite recent arrests of Hamas activists in the West Bank.

The Hamas pledge to continue attacks came after the Islamist group claimed responsibility for two shooting attacks on Israeli settlers in the past two days that left four people dead and two others wounded.

"We are under no illusions," Obama said this week. "Passions run deep ... there's a reason that the two-state solution has eluded previous generations -- this is extraordinarily complex and extraordinarily difficult."

Future details about the progress of negotiations will be limited, Mitchell told reporters. He noted that both the Israelis and the Palestinians have said that in order for negotiations to succeed, they "must be kept private" and "treated with the utmost sensitivity."

CNN's Ed Henry, Alan Silverleib, Hala Gorani and David Molko contributed to this report