The arrest added to jitters over the staging of European football's showcase event in France after last year's jihadist attacks in Paris.
Preparations for the long-anticipated event have also been marred by rolling strikes that have paralysed transport and torrential rain that caused flooding along the River Seine and forced tourist attractions including the Louvre Museum to close.
Ukraine's security service chief Vasyl Grytsak said the 25-year-old man intended to blow up "a Muslim mosque, a Jewish synagogue, tax collection organisations, transportation checkpoints and numerous other locations".
Video released by the SBU security service showed a man whose face was blurred out loading guns and other weapons into the back of a van.
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Ukrainian authorities said his arsenal included 125 kilogrammes of TNT and 100 detonators as well as bullets and grenade launchers.
The SBU said he was arrested on May 21 while trying to cross into Poland near the Ukrainian frontier town of Yagodyn.
The man has not been formally identified by French authorities, but he is believed to work for an agricultural cooperative inseminating cows in eastern France.
The employer of a man named in some media as the suspect told AFP he was an "exemplary employee".
A farmer who knows the man, but asked not to be named, said he regularly visited Ukraine.
"He told us he had a girlfriend in Ukraine and that he went to her place from time to time," the farmer said.
Washington has already warned US citizens about the risk of attacks on stadiums and fan zones.
France has mobilised 90,000 security personnel to secure areas with large numbers of supporters, but the increasingly anxious mood in France has cast a pall over a celebrated competition that unites the continent once every four years.