Conversion lines of the 279.16±0.02 and 401.27±0.05 keV transitions in Tl 203 have been studied at high resolution (≈ 0.07% in momentum) in the Chalk River iron-free π√ 2β- spectrometer . From the measured line intensities and the known total conversion coefficient (0.2262±0.0019) measured by Taylor the following conversion coefficients are deduced for the 279 keV M1+E2 transition: α K = 0.163±0.003, α L I = 0.0245±0.0004, α L II = 0.0162±0.0003, α L III = 0.0077±0.0003, α M = 0.0118±0.0003, α N+0 = 0.0035+0.0002. The measured L conversion ratios of the 401 keV transition show that it is pure M1 with < 2% E2, in accordance with earlier results. In order to test the accuracy of E2 theoretical L conversion coefficients, precise experimental line intensity ratios are reported for five pure E2 transitions in the region 76 ≦ Z ≦ 80. The trend of these ratios when compared with theory indicates that the true L II and L III coefficients may be a few percent lower than Sliv's theoretical predictions. The experiments were performed in order to study the nuclear structure effect on the internal conversion of the M1 part of the l forbidden 279 keV transition. The significance of the K and L conversion coefficients is discussed in detail. They are consistent with a mixing ratio of δ 2 = E2/M1 = 1.4±0.2 and a penetration parameter λ = | M e|/| M γ | ≈ 7 if one uses Sliv's conversion coefficients and Church and Weneser's predictions for the behaviour of β 1K, β 1L I , β 1L II and β 1L III as functions of γ for this transition. This value of γ is also consistent with the results of earlier experiments including those from γ-e − angular correlation studies when re-interpreted using the above value for δ 2 and the alteration in sign of the particle parameter modifying the interference term which has been pointed out recently by Geiger and Church et al. The effects of possible uncertainties in the present theoretical predictions are also considered and show that a solution with γ ≈ 73 cannot be conclusively rejected although with present knowledge it seems unlikely.