We present a new measurement of the top quark mass in ttbar events
in which both W bosons from top quarks decay into leptons (e nu, mu nu).  We
use events collected by the CDF experiment from ppbar collisions at sqrt s =
1.8 TeV at the Tevatron collider.  We measure a top quark mass of 167 +/-
10.3(stat) +/- 4.8(syst) GeV/c^2 from a sample of eight events.  We combine
this result with previous CDF measurements in other decay channels to obtain a
mass value of 176.0 +/- 6.5 GeV/c^2.