The deal gives Italy "the opportunity to restart with strength" and the Government the responsibility "to change the face of the country," Conte said.
Spain will receive the second largest sum, €140b, with just over half in grants and the remainder in loans.
The European Commission will raise capital against the EU Budget with richer countries effectively underwriting loans to fund spending in countries that would struggle to take on the debt on their national balance sheets.
"We did it! Europe is strong. Europe is united," said Charles Michel, the President of the European Council.
"These were, of course, difficult negotiations in very difficult times for all Europeans."
Leaders had argued over how money from the rescue fund should be paid out to the hardest hit countries in loans and grants. At the end, they agreed €390b of the rescue plan should be handouts and €360b in loans.
Leaders also agreed on a €1.074t Budget, the first to be agreed since Brexit and first without UK contributions since it joined the bloc in 1973, completing a €1.82t package.
Mark Rutte, the Prime Minister of the Netherlands, and his "frugal" allies of Austria, Sweden, Denmark and, at times, Finland blocked the original plan of €500b in grants and €250b of loans.
Germany and France had insisted at least €400b of the package must be grants to shield the fragile economies of southern Europe.
French President Emmanuel Macron, who accused Rutte of taking on Brexit Britain's intransigent role as the pair clashed during heated talks, called the compromise a "historic day for Europe".
Angela Merkel, the German Chancellor, said: "We have laid the financial foundations for the EU for the next seven years and came up with a response to this arguably biggest crisis of the EU."
The northern frugals did secure an "emergency brake" on the disbursement of money. Spain and Italy rejected Rutte's call for an outright veto on spending but other countries will be able to review and demand changes to their spending plans.
"Countries will know that there is an extra set of eyes, all of us in the EU sitting at that table, watching whether you are doing what you have committed to do," Rutte said.
