* Visit comes at low point for Hollande
* Discuss Middle East, family matters, environment
* Security tight after bomb explodes near French church
By Elizabeth Pineau and Philip Pullella
VATICAN CITY, Jan 24 (Reuters) - French President Francois Hollande met Pope Francis on Friday, making a delicate visit against the backdrop of undenied allegations of infidelity, low popularity ratings and clashes with Catholics over gay marriage.
Hollande and Francis held 35 minutes of private talks in the Apostolic Palace, which he now uses only for official visits after he moved into a modest apartment.
Security was beefed up and the exclusion zone around the Vatican was widened, a police source said, after a small makeshift explosive went off near a French church in Rome in the early hours and police received an anonymous call about two more bombs in the Vatican area. None were found.
Hollande seemed nervous as he was introduced to the pope and Francis, who does not speak French well, did not appear to give him a particularly warm welcome compared to other heads of state.
The Socialist president's decision to rush through the legalisation of same-sex marriage last year provoked several huge protests in Paris that were strongly supported by the Catholic hierarchy.
Some Catholics in France have accused the government of paying more attention to minority Muslims and Jews, visiting their communities and denouncing attacks on them while all but ignoring provocations against France's majority religion.
A Vatican statement said the two discussed several topics, including the family and bioethics.
Hollande told reporters afterwards that France's longstanding separation of church and state, known as "laicite," "guaranteed the rights of all faiths ... and allowed a debate with all religions, notably the Catholic Church".
Hollande said the debate on moral issues could include "the most serious and pressing matters facing our society" but gave no indication that the government would row back on gay marriage legislation or abortion rights or euthanasia.
Both Catholic and Protestant church leaders in France have denounced Hollande's announced plan to legalise assisted suicide this year.
The Catholic Church teaches that life begins at the moment of conception and ends at the moment of natural death. It is opposed to any form of euthanasia.
Many younger Catholics mobilised by the anti-gay marriage movement last year have kept their protest networks fresh. One group has collected 115,000 signatures on a petition sent to Pope Francis urging him to bring up their complaints in his meeting with Hollande.
Hollande aides said before the meeting he would "listen to what the pope has to say" about end-of-life policies but French Catholics doubt he will follow the pope's advice.
The two also discussed Syria, the Palestinian-Israeli conflict, the environment and the rights of Christians in the Middle East.
Hollande's visit came at a delicate time politically, with his popularity at record lows and France holding municipal elections in March.
Hollande has not denied reports of an affair with actress Julie Gayet that have put the president's private life in the spotlight.
The president, who did not take any questions from the media after his meeting with the pope, said Francis would be welcome to visit France whenever he wanted.
He has refused questions about his private life after celebrity magazine Closer published pictures of what it said was the president wearing a motorcycle helmet visiting Gayet for nocturnal trysts.
The French newspaper Le Monde on Friday ran a front page cartoon showing Hollande arriving at the Vatican on a motorcycle with two female passengers. The pope greets them, saying "Who am I to judge", his now-famous phrase on homosexuals. (Additional reporting by James Mackenzie in Rome and Tom Heneghan in Paris; Writing by Philip Pullella; Editing by Alison Williams)