Assistant District Attorney Robert Isdith said the neighbor, Irene Raymond, didn't want to pursue the case.
Brown pleaded not guilty to criminal contempt and her trial was slated to begin Tuesday.
Brown was issued the order of protection after pleading guilty in 2008 to menacing Raymond with her cellphone. The two had been in a fight over Brown blasting her car stereo outside their Brooklyn building.
Prosecutors said Brown, whose real name is Inga Marchand, violated the order in July 2010 by screaming at Raymond before bending over and mooning her.
The 31-year-old hip hop star smiled and hugged her attorneys when she heard that the case was dismissed.
Brown was a teenager when she broke onto the rap scene as a protege of Jay-Z.
The Associated Press contributed to this report.