The Parkhead club beat the transfer deadline on Tuesday to add the centre-back to the squad for the remainder of the season.
The agreement is a straight loan with no option to buy announced as part of the deal.
Arthur, 20, made his debut for Brentford earlier this season in a League Cup win over Aston Villa.
He has also represented England at under-20 level.
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Martin O'Neill said of the deal: "I'm really happy that we have brought Benjamin to the club, and I think he can be a very important addition to our squad during the remainder of the season.
"He is an excellent player, with real attributes, good height, strength and speed and I know he can only help and improve the squad across the number of games which we will face over the next few months.”
Arthur commented: “I am really honoured and happy to be joining Celtic. It is one of the world’s biggest clubs, so it’s a really exciting move for me.
“I know we have a quality squad with guys who have won so much already, and I can’t wait to join them and hopefully play my part in bringing more success to the club and to our fans.
“We have a lot of challenges ahead and I am ready for them and ready to do all I can to help the club achieve all it can.”