Plane-by-plane femtosecond laser fabricated apodized fiber Bragg gratings (FBG) are demonstrated for the first time, to the best of our knowledge. The method reported in this work provides a fully customizable and controlled inscription that can realize any desired apodized profile. By using this flexibility, we experimentally demonstrate four different apodization profiles (Gaussian, Hamming, New, Nuttall). These profiles were chosen to evaluate their performance with regard to the sidelobe suppression ratio (SLSR). Usually, a higher reflectivity of a grating fabricated with a femtosecond laser will result in a greater difficulty to achieve a controlled apodization profile due to the nature of the material modification. Therefore, the goal of this work is to fabricate high-reflectivity FBGs without sacrificing the SLSR and provide a direct comparison with apodized low-reflectivity FBGs. In our weak apodized FBGs, we also consider the background noise introduced during the femtosecond (fs)-laser inscription process which is fundamental when multiplexing FBGs within a narrow wavelength window.