The intensity of the 283.4-keV γ line in the decay of ¹³⁷Cs has been measured. The value obtained is used to determine the probability of the unique second-forbidden β transition of the 7/2⁺ → 1/2⁺ type: (8.7 ± 2.0) ×10^-4% of the number of decays; this value corresponds to log f₂^un t = 16.45 ± 0.12. The intensity of the possible E 5 transition at 378.3 keV is estimated to be < 0.9 × 10^-4%. The technique for determining the values of log f₂^un t in calculations of the tensor matrix elements in the Cartesian and spherical coordinate systems is discussed. The possible intensity of the unique second-forbidden β transition between the ground states of ¹²⁹I and ¹²⁹Xe is estimated to be I_β ≈ 0.1% of the number of decays of ¹²⁹I.