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The size of a ΔK ¼ 0 M1 excitation strength has been determined for the first time in a predominantly axially deformed even-even nucleus. It has been obtained from the observation of a rare K-mixing situation between two close-lying Jπ ¼ 1þ states of the nucleus 164Dy with components characterized by intrinsic projection quantum numbers K ¼ 0 and K ¼ 1. Nuclear resonance fluorescence induced by quasimonochromatic linearly polarized γ-ray beams provided evidence for K mixing of the 1þ states at 3159.1(3) and 3173.6(3) keV in excitation energy from their γ-decay branching ratios into the ground-state band. The ΔK ¼ 0 transition strength of BðM1; 0 þ 1 → 1 þ K¼0 Þ ¼ 0.008ð1Þμ2 N was inferred from a mixing analysis of their M1 transition rates into the ground-state band. It is in agreement with predictions from the quasiparticle phonon nuclear model. This determination represents first experimental information on the M1 excitation strength of a nuclear quantum state with a negative R-symmetry quantum number.