We have developed a fiber-based frequency comb system consisting of a simple mode-locked fiber laser and a backward pumping amplifier combined with a highly nonlinear fiber with a short zerodispersion wavelength. As a result, the signal to noise ratio of the obtained carrier-envelope-offset frequency beat is larger than 45 dB at a bandwidth of 100 kHz. Furthermore, we have succeeded in measuring the optical frequencies of a 1542-nm acetylene-stabilized laser and a 532-nm iodinestabilized Nd:YAG laser continuously for more than one week using the fiber-based comb system. The long-term measurement revealed that the frequency stability of the iodine-stabilized laser was 5.7 x 10(-15) with 100 000 s averaging.