Cullen, 32, ranks fourth among Carolina scorers this season with 40 points. He is tied for second with 28 assists and has played in all 60 games, the team said on its website (http://www.carolinahurricanes.com) on Friday.
Picard, 24, has recorded four goals and 11 assists (15 points) in 45 games with the Senators this season, his fifth in the NHL.
"Alexandre is a good, young defenseman who should become a regular part of the Hurricanes's blue line," said Carolina general manager Jim Rutherford.
The Senators also announced they had extended the contract of general manager Bryan Murray by one year, taking him through the 2010-11 season.
Murray has been general manager since June 2007.
"Bryan has proven he is the right man to lead this team," owner Eugene Melnyk told Ottawa's website (www.ottawasenators.com).
"Our team is now in a position to not only be competitive each season but also to contend for the Stanley Cup."
The Senators took over the Northeast Division lead with a 6-5 victory over the Washington Capitals on Thursday. They are tied third in the Eastern Conference with 74 points.
(Reporting by Gene Cherry, editing by Tony Jimenez)