Low-frequency high-precision laser interferometry is subject to excess laser-frequency-noise coupling via arm-length differences which is commonly mitigated by locking the frequency to a stable reference system. This approach is crucial to achieve picometer-level sensitivities in the 0.1-mHz to 1-Hz regime, where laser-frequency noise is usually high and couples into the measurement phase via arm-length mismatches in the interferometers. Here we describe the results achieved by frequency stabilizing an external cavity diode laser to a quasimonolithic unequal arm-length Mach-Zehnder interferometer readout at midfringe via balanced detection. We find this stabilization scheme to be an elegant solution combining a minimal number of optical components, no additional laser modulations, and relatively low-frequency-noise levels. The Mach-Zehnder interferometer is designed and constructed to minimize the influence of thermal couplings and to reduce undesired stray light using the optical simulation tool ifocad. We achieve frequency-noise levels below 100  Hz/Hz at 1 Hz and are able to demonstrate the LISA frequency prestabilization requirement of 300  Hz/Hz down to frequencies of 100 mHz by beating the stabilized laser with an iodine-locked reference.