Consider the simple random walk on the n-cycle ℤn. For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695 have shown that the log-Sobolev constant α is of the same order as the spectral gap λ. However the exact value of α is not known for n>4. (For n = 2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that α is 1/2. For n = 3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that α = 1/2log2 < λ/2 = 0.75. For n = 4, the fact that α = 1/2 follows from n = 2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if n ≥ 4 is even, then the log-Sobolev constant and the spectral gap satisfy α = λ/2. This implies that α = 1/2(1 - cos 2π/n) when n is even and n ≥ 4.