Here's an outline of Europe's two biggest economies' proposals for a post-Brexit Europe:
A record influx of migrants, many fleeing war in Syria and Iraq, has become a defining issue for politics in many EU nations.
Merkel is facing down a domestic rebellion over her liberal policy that led to the arrival of more than a million asylum seekers since 2015, while Italy last week sparked an outcry by refusing aid ships carrying rescued migrants to dock in its ports.
Coming to Merkel's aid, Macron said that Paris and Berlin had agreed that EU countries must be able to turn back at their borders all asylum seekers who previously registered in another member country -- usually their Mediterranean first port of call.
Merkel, asked earlier what Berlin can offer other EU countries in return for such cooperation, cited the EU-Turkey deal in which the bloc paid Ankara for housing and schooling refugees.
They are also calling for a European Asylum Office that harmonizes practices across the bloc, and which will be responsible for "asylum procedures at external borders".
Meanwhile, Paris and Berlin also vowed steps to better protect the EU's outside borders by boosting the Frontex agency, to fight human traffickers and create a fairer system of burden-sharing within the bloc.
The German leader had initially championed a quota system to distribute migrants across the bloc, but this has been torpedoed by several central European nations.
SEE ALSO: Just how 'open' are Germany's borders?
Macron's ambitious ideas for a eurozone finance minister or parliament have been shot down by Berlin.
But the French leader won Merkel's backing for a eurozone budget which will be used to finance things like projects on innovation and education in poorer EU countries.
The eurozone budget will have its own governance structure, and be a "real budget with annual revenues and spending", said Macron, adding that Paris and Berlin hoped to have it in place by 2021.
He would however not be drawn into giving a figure for the budget, saying that its size would be discussed with other members of the bloc.
How it would be funded was also up for discussion with other eurozone members, said Merkel, suggesting that it could involve regular transfers made by individual countries or a tax on financial transactions.
Macron had previously called for a budget of several hundred billion, while Merkel has spoken of some tens of billions of euros.
Beyond that, in order to better cope with future crises, the two leaders agreed to expand the remit of the current European Stability Mechanism (ESM).
The twin EU engines now want the ESM to become a European version of the International Monetary Fund that could provide emergency loans to countries that fall victim to crises not linked to their debt levels.
In the face of US President Donald Trump's "America First" policy, a traditionally pacifist and Atlanticist post-war Germany has drawn closer to France's call for a sovereign Europe, including in the military aspect.
Merkel backs Macron's idea for an European intervention initiative, with armies to collaborate before eventually forming a common force.
But France wants this force outside the framework of the EU, to allow Britain to join in even after it leaves the bloc.
Germany prefers it to remain within the EU.
As a compromise, both sides agreed for the force to be "linked as closely as possible" with the EU's framework.
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