Councillor Cox, 52, who represents Westbourne ward, won his seat in a by-election in December 2011.
He said that it would be inappropriate to contest both elections – for a seat in Parliament and on the council – next May.
So he will stand down when both elections are held on Thursday 7 May 2015.
His Conservative colleague Denise Cobb, currently the deputy mayor, is standing again and another candidate is being sought.
In a newsletter to people living in his ward Councillor Cox wrote: “Denise and I have to begin our midsummer newsletter by referring to the elections which will be taking place next year.
“Mike Weatherley, the MP for Hove, has announced that he will be standing down next year.
“Mike has fought a brave and largely private battle with cancer whilst continuing to represent us at Westminster.
“We are delighted that he was recently given the all-clear, but fully understand his decision to reassess things following this experience.
“Many of you will have heard by now that I have been selected by the local party to contest the Hove and Portslade seat as the Conservative candidate in 2015.
“For the first time for many years, the parliamentary and city council elections will take place on the same day in May 2015.
“It would be inappropriate for me to contest both elections, and therefore I will not be standing as a councillor for Westbourne next year – although hopefully I will be able to continue to represent Westbourne at Westminster rather than in the town hall.
“Obviously this will create a vacancy for a Conservative candidate for Westbourne ward.
“We are keen to open the opportunity to stand to as many people as possible, so if you are interested in putting yourself forward do get in touch and I will happily explain what is involved.”
LIKE WHAT WE DO? HELP US TO DO MORE OF IT BY DONATING HERE.