LONDON — With less than 18 months to go before Britain is scheduled to leave the European Union, and with British businesses growing increasingly concerned about the future, this was the moment Prime Minister Theresa May hoped for a breakthrough in the paralyzed negotiations on the process known as Brexit.

Instead, she is fighting a tense battle to stop the talks from collapsing.

After phone calls to Berlin and Paris, and the first of two visits to Brussels in three days, the most Mrs. May accomplished on Tuesday was a promise to keep talking. However, this dialogue does not yet include the subjects Mrs. May wants to discuss: a transition deal to prevent an economically damaging “cliff edge” Brexit and future trade relations.

This week, she is to make her pitch to European Union leaders at a summit meeting in Brussels, but the main discussion of Britain’s departure from the bloc will happen after Mrs. May has left the room.

Both at home and abroad, Britain’s prime minister is hamstrung by her political fragility. She is constantly forced to mediate between warring factions in her cabinet, some of whom want a quick, clean break with the bloc, while others fret about protecting the economy. She must also fend off doubts on the Continent about her ability to deliver a deal, even if one is agreed to.