We demonstrate the use of a plasma mirror to obtain 60-fs 10-TW laser pulses with a temporal contrast of 10(8) on a nanosecond time scale and 10(6) on a picosecond time scale, and we use these high-contrast pulses to generate high harmonics by nonlinear reflection on a plasma with a steep electronic density gradient. Well-collimated harmonics up to 20th order are observed for a laser intensity of approximately equal to 3 x 10(17) W/cm2, whereas no harmonics are obtained without the plasma mirror.