Shin Bet said Franck received the weapons from a Palestinian resident of the Gaza Strip employed at the French Cultural Centre there. He then transferred the weapons to the West Bank, where they were sold to arms dealers, the security agency said.
"We take this case very seriously and are in very close contact with the Israeli authorities on this case," said a spokesman for the French Embassy in Tel Aviv.
Franck reportedly has been held since February 15.
In connection with the smuggling ring, Israel said it also arrested a Palestinian resident of East Jerusalem who works as a security guard at the French Consulate in Jerusalem, as well as several Palestinians from Gaza living illegally - under Israeli law - in the West Bank. Six of them will be indicted by the Southern District Court, the Shin Bet statement said.
Some of the suspects were also involved in smuggling money from the Gaza Strip to the West Bank, Shin Bet alleged.
Gaza has been subject to strict restrictions on movement and trade since Hamas took control of the enclave in 2007. Since then, Israel and Hamas - designated a terrorist organisation by Israel, the United States and the European Union - have fought three wars.