A tapered semiconductor amplifier is injection seeded by a femtosecond optical frequency comb at $780\mathrm{nm}$ from a mode-locked Ti:sapphire laser. Energy gains of more than $17\mathrm{dB}$ $(12\mathrm{dB}$ ) are obtained for $1\mathrm{mW}$ $\left(20\mathrm{mW}\right)$ of average input power when the input pulses are stretched into the picosecond range. A spectral window of supercontinuum light generated in a photonic fiber has also been amplified. Interferometric measurements show sub-Hertz linewidths for a heterodyne beat between the input and amplified comb components, yielding no detectable phase-noise degradation under amplication. These amplifiers can be used to boost the infrared power in f-to- $2f$ interferometers used to determine the carrier-to-envelope offset frequency, with clear advantages for stabilization of octave-spanning femtosecond lasers and other supercontinuum light sources.