I mean, there are two universities which refer to themselves as "University of Barcelona":

1. Universitat de Barcelona (http://www.ub.edu/homeub/welcome.html), public, founded in 1450. Ranked at #4 in all of Spain. UB  ranked as #6 according to El Mundo newspaper (2007/2008) and #3 according to Gaceta Universitaria (2002)
 
2. Universitat Autonoma de Barcelona (http://www.uab.es), public, founded in 1969. UAB ranked as #3 according to El Mundo newspaper (2007/2008) and #4 according to Gaceta Universitaria (2002)

Source for ranking: http://es.wikipedia.org/wiki/Ranking_de_universidades_espa%C3%B1olas