On July 23 last year, a security guard for the Israeli embassy shot dead a Jordanian worker who had come to an apartment to install furniture and who stabbed him in the back with a screwdriver, according to the Israeli foreign ministry.
A second Jordanian, the apartment landlord, was also killed - apparently by accident.
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Jordan's government spokesperson Mohammed Momani said Thursday Amman had "received an official memo from the Israeli foreign ministry expressing the Israeli government's strong sorrow and regret over the Israeli embassy incident."
It had also apologised for the killing of a Jordanian judge by an Israeli soldier at the countries' border in 2014 and pledged to compensate all three families, Momani said in statements carried by the official Petra news agency.
A statement Thursday from the office of Israeli Prime Minister Benjamin Netanyahu said Israel "attaches great importance to its strategic relations with Jordan and the two states will act to advance cooperation between them and to strengthen" a 1994 peace treaty between them.
It made no mention of an apology or compensation.
The embassy guard was briefly questioned by investigators in Jordan but was allowed to return Israel along with the rest of the embassy staff, on the grounds he had diplomatic immunity.
He received a hero's welcome from Netanyahu, sparking widespread anger in Jordan.
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Amman later said it would not allow the embassy staff to return until Israel opened a serious investigation and offered an apology.
Momani said Israel's memo had met all the conditions the government had set for the Israeli ambassador to return to Amman.
"The government contacted the families of all three martyrs who agreed to accept the apology and compensation," he added.
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