Under the emerging agreement, Hamas would hand over responsibilities of governing Gaza to the West Bank-based government of Prime Minister Rami Hamdallah.
Al-Ahmad, who is also the head of Fatah delegation for the Cairo talks, said what was agreed upon two weeks ago under the auspices of Egypt is empowering the legitimate government to operate in accordance with its powers in accordance with the Basic Law in effect in the West Bank.
Hamas and Fatah would return to Cairo in early December to assess implementation of the agreement, the official said.
Egypt has previously attempted to help Hamas and Fatah create a united government for Gaza and the West Bank.
Al-Ahmed told reporters after the ceremony that there has been "full agreement" to empower a national unity government to assume its full authorities in Gaza. Abbas has said he would not take over Gaza until Hamas gave up its weapons.
"Any reconciliation between the Palestinian Authority and Hamas must meet the Quartet's conditions - accepting worldwide agreements, recognizing Israel, and disarming Hamas", Israeli officials said in a statement after the deal was announced.
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He added that sanctions taken by Abbas against Gaza will also soon be lifted.
The Western-backed mainstream Fatah party lost control of Gaza to Hamas in fighting in 2007.
The readout of the Secretary-General's phone call comes as media outlets are reporting that Egyptian-facilitated talks in Cairo have led to a breakthrough in the talks among Palestinian parties on administration in the Gaza Strip.
A split between Palestine's two major forces, Fatah and Hamas, occurred after the parliamentary elections in Palestine in 2006 when Hamas won.
The two sides had been meeting in the Egyptian capital since Tuesday, with the aim of ending a crippling decade-old split between the rival factions.
Struggling with the fallout from the border blockade, Hamas has found it increasingly hard to govern or provide basic services, such as electricity, to Gaza's people.