France is in a state of emergency after militants killed 130 people in a series of coordinated assaults across Paris in November.
The French, United States and other governments have warned that militants may target the tournament.
Ukraine's state security service also said on Monday a French citizen detained on the border with Poland had been planning attacks on Jewish and Muslim places of worship in France to coincide with the soccer championship.
Ukraine arrests Frenchman 'planning Euro 2016 attacks'
"Today we are as well prepared as we can be," police chief Michel Cadot told a news conference.
LIVE #Parigi Conferenza stampa prefetto Michel Cadot: "Ci prepariamo da un anno a #EURO2016" Foto @Annagmilan pic.twitter.com/QWgFvlsTEW
— Rai Radio1 (@Radio1Rai) June 6, 2016
Nationwide, more than 90,000 police, soldiers and private security agents will be deployed to ensure the safety of the June 10- July 10 tournament.
But officers say they are tired and stressed after contending with two militant attacks in a year and regular, sometimes violent street protests.
Cadot said he would be receiving more than 3,000 additional security agents to support the planned 10,000-strong force.
These, he said, would include fifteen 100-strong mobile units.
"To take on this heavy responsibility that is Euro 2016, we needed extra resources. The minister gave them to me. So from this point of view I got the desired response," Cadot said, in reference to his May 26 letter voicing concerns over force size.
Islamic State militants have threatened a campaign of bomb attacks on large crowds in France, the head of the DGSI internal intelligence said on May 19.
About 1.5 million foreign fans are expected for 51 soccer matches involving 24 teams in games played at 10 stadiums across France.
Paris attacks suspect Abdeslam transferred to France
There will also be fan zones for crowds watching games on giant TV screens in cities, including at the Eiffel Tower, where up to 100,000 people are expected on match days.
Security experts and police officials have said protecting those areas will be the biggest challenge due to a lack of resources.
Cadot last month told the interior ministry that the Eiffel Tower fan zone should be closed during games played at stadiums in the city and its outskirts to avoid stretching his force too thin.
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