We present measurements of the process p[overline p]-->WZ+X-->[script-l][prime]nu[script-l][prime][script-l][overline [script-l]] at sqrt(s)=1.96 TeV, where [script-l] and [script-l][prime] are electrons or muons. Using 1 fb-1 of data from the D0 experiment, we observe 13 candidates with an expected background of 4.5±0.6 events and measure a cross section sigma(WZ)=2.7-1.3+1.7 pb. From the number of observed events and the Z boson transverse momentum distribution, we limit the trilinear WWZ gauge couplings to -0.17<=lambdaZ<=0.21(DeltakappaZ=0) at the 95% C.L. for a form factor scale Lambda=2 TeV. Further, assuming that Deltag1Z=DeltakappaZ, we find -0.12<=DeltakappaZ<=0.29(lambdaZ=0) at the 95% C.L. These are the most restrictive limits on the WWZ couplings available to date.