Gloucester expect him to be fit within six to eight weeks so should be back in action during March.
"We were pretty sure that Alex had fractured his arm on Saturday and we took every measure we could to move as quickly as possible to start his recovery process," said head coach Dean Ryan. "He is obviously a significant member of our squad and I think any side would miss such a key individual at this stage of the season.
"But we have been in this situation before and will manage. Alex is a notoriously quick healer and we will give him all the support he needs until he is fit again."
Brown has made 149 Guinness Premiership appearances throughout his career and has already made 15 appearances this season. He has three England caps.