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We determine the critical couplings for the deconfinement phase transition in SU(2) gauge theory on N(tau) x N(sigma)3 lattices with N(tau) = 8 and 16 and N(sigma) varying between 16 and 48. A comparison with string tension data shows scaling of the ratio T(c)/square-root sigma in the entire coupling regime beta = 2.30-2.75, while the individual quantities still exhibit large scaling violations. We find T(c)/square-root sigma = 0.69(2). We also discuss in detail the extrapolation of T(c)/LAMBDA(MSBAR) and square-root sigma/LAMBDA(MSBAR) to the continuum limit. Our result, which is consistent with the above ratio, is T(c)/LAMBDA(MSBAR) = 1.23(11) and square-root sigma/LAMBDA(MSBAR) = 1.79(12). We also comment upon corresponding results for SU(3) gauge theory and four-flavour QCD.