We report measurements of the decays B(+)-->phiphiK(+) and B(0)-->phiphiK(0) using a sample of 231 x 10(6) BB pairs collected with the BABAR detector at the PEP-II asymmetric-energy B factory at the Stanford Linear Accelerator Center. The branching fractions are measured to be B(B(+)-->phiphiK(+))=(7.5+/-1.0(stat)+/-0.7(syst)) x 10(-6) and B(B(0)-->phiphiK(0))=(4.1(-1.4)(+1.7)(stat)+/-0.4(syst)) x 10(-6) for a phiphi invariant mass below 2.85 GeV/c(2).