Eclogite, the high-pressure equivalent of basalt, is common in ultrahigh-pressure metamorphic terranes and constitutes the oceanic crust after it is subducted into the mantle. We present here first measurements of the rheology of eclogite and its constituent minerals, garnet and omphacite. Eclogite with approximately equal amounts of these two minerals has the same strength as harzburgite, whereas pure omphacite is much weaker and pure garnetite is much stronger. The strength variation of different mixtures of the two phases is quantitatively consistent with theoretical predictions for a two-phase material. The strengthening effect of garnet should increase the viscosity of the lower part of the mantle transition zone, but it probably is not sufficient to allow delamination of the oceanic crust from the underlying mantle during subduction.