We have measured the mechanical strength (σ) of pure water ices V and VI under steady state deformation conditions. Constant displacement rate compressional tests were conducted in a gas apparatus at confining pressures from 400 < P < 800 MPa, temperatures from 209 < T < 270 K, and strain rates 7 × 10−7 < < 7 × 10−4 s−1. Most of the results fit to an empirical flow law of the form = A σn exp {−(E* + PV*)/RT}, where the four material constants A, n, E*, and V* are (for in inverse seconds and P and σ in megapascals) 1023.0, 6.0, 136 kJ/mol, and 29 cm3/mol, respectively, for ice V; and 106.7, 4.5, 66 kJ/mol, and 11 cm3/mol, respectively, for ice VI. Ice VI may weaken to a mechanism of higher E* at T > 250 K. Ices V and VI are thus rheologically distinct but by coincidence have approximately the same strength under the conditions chosen for these experiments. To avoid misidentification, these tests are therefore accompanied by careful observations of the occurrences and characteristics of phase changes. One sample each of ice V and VI was quenched at pressure to metastably retain the high‐pressure phase and the acquired deformation microstructures; X ray diffraction analysis of these samples confirmed the phase identification. Surface replicas of the deformed and quenched samples suggest that ice V probably deforms largely by dislocation creep, while ice VI deforms by a more complicated process involving substantial grain size reduction through recrystallization.