We describe the operation of a laser cooled (87)Rb frequency standard and present a new measurement of the (87)Rb ground state hyperfine frequency with a relative accuracy of 2.4x10(-15), by comparison with a Cs fountain primary standard. The measured frequency is 6 834 682 610.904 333(17) Hz. An evaluation of the frequency shift induced by cold collisions gives Deltanu/nu(Rb)=(-7.2+/-20)x10(-24) n , where n is the average atomic density in cm(-3). With our present 1 sigma uncertainty of 10(-15), this measurement is still compatible with 0 and about 300 times smaller than for (133)Cs. We also report a test of a possible variation of the fine structure constant at the level of 2.7x10(-14) yr(-1), comparing Rb and Cs cold atom fountains.