Robert Griffin III Got His New Chick Tatted On His Arm

Seems RG3 is ALL IN with his new girlfriend ... 'cause he got her name TATTOOED ON HIS ARM!!

We're told the two hit up a tattoo parlor in Cleveland last month -- both got brand new ink on their forearms.

Robert's tat reads ... GRETE SADEIKO.

Boom.

Grete is the 23-year-old Florida State track star who has apparently been dating the Browns QB for at least a month.

By the way, Griffin is STILL MARRIED to his wife, the mother of his child ... though they are recently separated.

As for Grete's tattoo -- hard to see what it says. Wonder if it's his name?