The low-temperature ultrasonic dispersion in the transverse (C-11 - C-12)/2 mode of PrMg3 with the non-Kramers-doublet ground state has been investigated in temperatures down to 20 mK. The dispersion is caused by a thermally activated motions characterized by an Arrhenius-type relaxation time of tau = tau(0) exp(E/k(B)T) with low activation energy, E = 500 mK, and slow attempt time, tau(0) = 0 8.5 x 10(-10) s. The coupling of the electric quadrupoles of the non-Kramers-doublet ground state to transverse lattice vibrations leads to a vibronic ground state with dissipation. The vibronic state in PrMg3 releases the entropy of k(B) ln2 with lowering temperature across the activation energy. A Kondo-like singlet state due to the binding of the non-Kramers doublet to the lattice vibrations appears at low temperatures far below the activation energy.