We demonstrate a slow-light augmented unbalanced Mach-Zehnder interferometer (MZI) which can be used to enhance very significantly the sensitivity of measuring the frequency shift in a laser. The factor of enhancement depends on the group index of the slow-light medium, the degree of imbalance between the physical lengths of the two arms of the MZI, and the spectral width of the laser. For a laser with a quantum noise limited spectral width, the group index has to be larger than the finesse of the laser cavity in order to achieve enhancement in measurement sensitivity. For the reported results, slow-light effect is produced by employing electro-magnetically induced transparency via coherent population trapping in a paraffin coated vapor cell of Rb atoms, with a maximum group index of ~2170. The maximum enhancement factor realized is ~70. This differs from the theoretically expected value of ~183 by a factor of ~2.6. This discrepancy can be attributed to effects of unidentified excess noise, and the fact that the test laser spectral width may not the quantum noise limited. Much larger values can potentially be obtained by modifying the apparatus, and using cold atoms for producing the slow-light effect. The sensitivity of any sensor that relies on measuring the frequency shift of a laser can be enhanced substantially using this technique. These include, but are not limited to, gyroscopes and accelerometers based on a conventional ring laser or a superluminal ring laser, and detectors for virialized ultra-light field dark matter.