Methane clathrate hydrate (structure I) is found to be very strong, based on laboratory triaxial deformation experiments we have carried out on samples of synthetic, high-purity, polycrystalline material. Samples were deformed in compressional creep tests (i.e., constant applied stress, σ), at conditions of confining pressure P = 50 and 100 MPa, strain rate 4.5 ¿ 10-8 ≤ $dot{varepsilon}$ ≤ 4.3 ¿ 10-4 s-1, temperature 260 ≤ T ≤ 287 K, and internal methane pressure 10 ≤ PCH4 ≤ 15 MPa. At steady state, typically reached in a few percent strain, methane hydrate exhibited strength that was far higher than expected on the basis of published work. In terms of the standard high-temperature creep law, $dot{varepsilon}$ = Aσne-(E*+PV*)/RT the rheology is described by the constants A = 108.55 MPa-n s-1, n = 2.2, E* = 90,000 J mol-1, and V* = 19 cm3 mol-1. For comparison, at temperatures just below the ice point, methane hydrate at a given strain rate is over 20 times stronger than ice, and the contrast increases at lower temperatures. The possible occurrence of syntectonic dissociation of methane hydrate to methane plus free water in these experiments suggests that the high strength measured here may be only a lower bound. On Earth, high strength in hydrate-bearing formations implies higher energy release upon decomposition and subsequent failure. In the outer solar system, if Titan has a 100-km-thick near-surface layer of high-strength, low-thermal conductivity methane hydrate as has been suggested, its interior is likely to be considerably warmer than previously expected.