  Weak measurement (WM) with state pre- and post-selection can amplify
otherwise undetectable small signals and thus promise great potentials in
precision measurements. Although frequency measurements offer the hitherto
highest precision owing to stable narrow atomic transitions, it remains a
long-standing interest to develop new schemes to further escalate their
performance. Here, we propose and demonstrate a WM-enhanced spectroscopy
technique which is capable of narrowing the resonance to 0.1 Hz in a
room-temperature atomic vapor cell. Potential of this technique for precision
measurement is demonstrated through weak magnetic field sensing. By judiciously
pre- and post-selecting frequency-modulated input and output optical states in
a nearly-orthogonal manner, a sensitivity of $\text{7 fT/}\sqrt{\text{Hz}}$
near DC is achieved, using only one laser beam of $\text{7 }\text{\mu W}$
power. Additionally, our results extend the WM framework to a non-Hermitian
Hamiltonian, and shed new light in metrology and bio-magnetic field sensing
applications.
