The last round of talks in Geneva in September aimed at ending Yemen's nearly four-year-old war were cancelled after three days of waiting for a Houthi delegation, which failed to arrive.
The almost four-year-old conflict has killed more than 10,000 people and caused a humanitarian disaster that could potentially threaten millions of lives.
The world must act now to end the war in Yemen and save millions of people from the brink of starvation, the man who is trying to broker a peace deal has told Sky News.
He said all sides to the conflict, including the Houthis, the Yemeni government, Saudi Arabia and the United Arab Emirates - another main player in the coalition - as well as the worldwide community want the fighting to end. They also point to the worldwide political pressure applied on Saudi Arabia by its allies, particularly after the killing of the journalist...
Setting out a timeline for peace, he said he hoped all sides would meet in Sweden "within the next few weeks" in what he said would be an "extraordinarily important first round" more than two years since the last time dialogue took place in Kuwait, only to break down.
"We must seize this positive worldwide momentum on Yemen", he told the U.N.'s most powerful body. "This is an opportunity at a crucial moment to pursue a comprehensive and inclusive political settlement to the conflict".
Griffiths said preparatory issues for the meeting are close to being resolved and he has sent the parties his "vision" for "U.N. -led, inclusive Yemeni negotiations to end the war and restart a political transition process".
The Houthis now rule over most of Yemen's population, while the exiled government controls a section of the south. Riyadh has waged war on its southern neighbor since March 2015, to overthrow the Houthi-led government, which Saudi Arabia accuses of being a proxy of Iran.
Saudi-led airstrikes have hit schools, hospitals and wedding parties and killed thousands of Yemeni civilians. The Houthis have fired long-range missiles into Saudi Arabia and targeted vessels in the Red Sea.
"He said 'every day about 50 children are brought to us. The situation in Al Hudaydah as a whole has escalated even stronger than before", the minister said.
He said that will require scaling up assistance to help 12 million Yemenis, which would require about $150 million to $160 million a month. And it would require a cash injection of $200 million a month into Yemen's economy to stabilize its currency and stave off economic collapse, he said.
"That port has got to be protected at all costs", Beasley said. "Children are already dying".
"You can not solve the humanitarian crisis in Yemen today with humanitarian response alone".
The Yemeni civil society representative, Rasha Jarhum, also spoke at the session, and called for aid in all aspects.
The UK on Monday will present a draft resolution to the Security Council to address the crisis in Yemen, Ambassador Karen Pierce said, as diplomatic efforts to end the war gathered pace.