We have discovered an interference which results in the vanishing of some electric dipole transition amplitudes. A transition-matrix element T between certain S and P sublevels with the same value of MF vanishes when an applied magnetic field satisfies Xs+Xp=-2MF, where X=μBgJBA, and A is the hfs constant. There must be an avoided crossing between the two states of the same MF which can only occur when one of the hyperfine manifolds has an inverted Zeeman effect. These conditions imply MF<0 and I>J. We observed the vanishing of T in 23Na for one of the ΔMF=0 optical transitions of the 3S12(F=1)-->3P12(F=1) manifold. Absorption of cw laser light was monitored by observing total fluorescence perpendicular to both the atomic and laser beams, and absorption vanished near an applied field of 155 G. T vanishes for a large number of cases including rf and microwave frequencies and is a rather general consequence of angular momentum selection rules and perturbation theory. This phenomenon may have application in optical pumping, Lamb-shift measurements, and atomic weak neutral-current experiments.