The midfielder has called time on his playing career after spending the last two years of his career at the Tameside Stadium, leading the club through one of its most successful periods in history.
He has decided to retire in a bid to allow more time to his business and also his family.
Brown has also represented Hyde United, Chester, Droylsden and a host of other local teams during his successful career.
Manager John Flanagan has thanked Brown for his dedication and commitment to Curzon during his time at the club. He said: "I would like to thank Alex for a fantastic two seasons with the club.
"With him, we have gained promotion from the Evo-Stik Northern Premier Division via the play-offs and established ourselves in the National League North.
"Alex has been a pleasure to manage. He is a fantastic lad, a great leader and a true example for everybody.
"He is a popular lad and a real Rolls Royce of a footballer, always playing to the highest quality.
"Thank you very much, Alex. All the very best for the future."
The first-team captaincy will now be resumed by goalkeeper Hakan Burton, who steps up from his position as vice-captain.
A new vice-captain will be announced in the near future.