A group of 49 migrants that were left stranded on board two NGO-operated rescue vessels disembarked in Malta on Wednesday afternoon.
Italian Deputy Premier Matteo Salvini has slammed a deal on rescued migrants that was brokered by the European Union, saying it will encourage human traffickers.
Early in the stalemate, Mr Muscat had also struck a defiant note, saying his tiny island nation would not let the 49 people set foot on Malta unless the deal included an agreement for fellow European Union countries to take numerous 249 migrants who Maltese military ships had rescued in its search-and-rescue area in late December.
Malta has been pushing other European countries to take in around 250 migrants now in its care before it would allow the two rescue boats to disembark.More news: North Korean leader visits econ-tech zone in Beijing before heading home
Malta said another 78 migrants would be allowed to stay in Malta, and 44 Bangladeshi migrants would be sent back to their country.
Muscat, who has previously complained that his country has to bear an unfair share of migrant numbers, insisted Malta had "never closed its ports and it is still a safe port".
The crew on the migrant rescue vessel had said the situation on the ship was "dire" after more than two weeks of those rescued stranded at sea.
Germany expects to take in 60 migrants who were already in Malta or are about to arrive there under an agreement to defuse the latest standoff over rescue ships in the Mediterranean Sea.
Sea-Watch said that the migrants finally have a "safe haven" after three weeks at sea.More news: Chandler Parsons, Memphis Grizzlies agree to part ways
The migrants will first be transferred to Maltese army patrol boats before arriving on the European Union island nation.
In the meantime, those aboard the Sea Watch and Sea Eye - who include a baby and several children - were celebrating the end of 19 days at sea.
Thanking civil society for its support, Sea Watch commented: "Disembarkation can not be subject to negotiations between states at the expense of people".
The disembarkation marks the end of an nearly three-week ordeal, which saw European member states refuse the disembarkation of the rescued people until a relocation agreement was reached.
"I am and I will remain absolutely against new [migrant] arrivals in Italy", he said on Twitter.More news: Fox News host Shep Smith wasted no time fact-checking Trump's address
Some 113,482 migrants crossed the Mediterranean to reach Europe past year, according to the United Nations refugee agency, which said 2,262 people lost their lives or went missing making the perilous journey.
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