The intrinsic thermal noise in optical fibers represents the ultimate limit for fiber-based systems. However, at infrasonic frequencies, the spectral behavior of the intrinsic thermal noise is still unclear. In this letter, we present measurements of the fundamental thermal noise in optical fibers that are obtained using a balanced fiber Michelson interferometer. When an ultra-stable laser is used as the laser source and other noise sources are carefully controlled, the 1/f spectral density of the thermal noise is observed down to infrasonic frequencies, and the measured magnitude is consistent with the results of theoretical predictions at frequencies over the range from 0.2 Hz to 20 kHz. Moreover, as observed experimentally, the level of the 1/f thermal noise can be reduced by changing the coatings of the optical fibers. This therefore indicates one possible way to reduce thermal noise in optical fibers at low Fourier frequencies. Finally, the inconsistency between the experimental data and the existing theory for thermomechanical noise is discussed.