The two-day Brussels summit marks the debut of French President Emmanuel Macron, the figurehead of a renewed confidence among the remaining 27 states that Britain's withdrawal can be a fresh start.
But talks on issues including post-Brexit defence plans risk being overshadowed by concerns that a disastrous election has left May so enfeebled that Brexit negotiations will be hampered.
"There is an enormous insecurity among the Europeans: how long will she last? Has she got the majority to deliver?" a senior EU official said.
Over dinner, May is expected to fill in some of the blanks for the other EU leaders on Brexit.
It will be their first meeting since her Conservative party unexpectedly lost its majority in a June 8 election, leaving her in charge of a so-called "zombie government".
Britain's shock referendum vote to leave the EU was a year ago on Friday, and the country remains in a dark national mood after a string of terror attacks and a deadly tower block blaze.
During the dinner May will "outline some principles of the UK's paper on citizens rights which will be published at the beginning of next week," the spokesman said.
The EU has made a priority of the rights of three million European citizens living in Britain, plus a million Britons resident in Europe.
At the first formal Brexit negotiations Monday, Britain accepted the EU's timetable that the exit bill, citizens' rights and the Northern Ireland border be settled before its request for a free trade deal be considered.
"We believe that the warming-up round of last Monday did create a positive atmosphere ... I don't think that May will want to shatter that understanding," said one EU diplomatic source, who asked not to named.
After her comments, May will leave the room for the remaining 27 EU member states to discuss what she has told them, and the future relocation of key EU agencies from London.
EU President Donald Tusk said the bloc appeared to have survived the worst of the anti-EU sentiment which drove Britain's shock vote to leave exactly a year to the day on Friday.
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