The atomic beam magnetic-resonance method with separated oscillatory fields has been used to measure the hyperfine structure separation and magnetic dipole moment of the isotopes Rb85 and 18.6-day Rb86 in the S122 electronic ground state. Observation of the separation of a ΔF=±1 doublet in the intermediate field region gives the value of the moment; the minimum value of the mean doublet frequency gives the value of Δν. Observation of another ΔF=±1 doublet in low field also yields a value for Δν. Results obtained for Rb85 are in good agreement with previously published values and indicate that transition frequencies calculated from the Breit-Rabi equation agree with experiment to at least one part per million. For Rb86 the following values are obtained for the S122 ground state: Δν=3946.883(2) Mc/sec, gI=−4.590(4)×10−4, and μI=−1.6856(14) nm (without diamagnetic correction). The hyperfine-structure anomaly arises in part from the difference of the volume distribution of nuclear magnetism in the two nuclei and is defined as the deviation from equality of the ratio of the hyperfine-splitting factors of two isotopes to the ratio of their nuclear g factors. For these two isotopes its value is found to be Δ8685=0.17(9)%. The Bohr-Weisskopf theory of the hfs anomaly is applied to these isotopes with calculations based on a single-particle model with varying distributions of spin and orbital contributions to the magnetic moment.