(Updates death toll, number rescued)

ROME, Sept 5 (Reuters) - Fifteen bodies were recovered and more than 2,700 boat migrants rescued off the coast of Libya on Monday, the Italian coastguard said, in another day of mass departures from north Africa.

Italy's navy and coastguard, ships patrolling on a European Union anti-smuggling mission, vessels run by humanitarian groups, and a commercial tug boat aided in the rescues.

Earlier in the day, the Italian Navy said six bodies had been found after migrants fell out of a leaking rubber boat. The coastguard gave no further details.

The migrants were saved from 19 dangerously overcrowded rubber boats and four small boats, the coastguard said. People smugglers operate freely in Libya, cashing in on migrants desperate to reach Europe.

Last week calmer seas and Libya's lawlessness opened the way for smugglers to ship 13,000 migrants across the Mediterranean Sea in just four days.

Europe's worst migrant crisis since World War Two is now focused on Italy, at Europe's southern frontier, where some 93,000 people had arrived by the end of August, according to Italy's Interior Ministry.

The death toll on the route from North Africa to Italy has jumped to one migrant for every 42 making the crossing, compared to one in every 52 last year, a U.N. refugee agency spokesman said last week. (Reporting by Isla Binnie and Steve Scherer; Editing by Catherine Evans)