  Ultra-sensitive measurement of the magneto-optical rotation, due to
interaction of linearly-polarized light passing through room-temperature Rb 85
atoms, in response to change in longitudinal magnetic field (\delta B_z ) is
demonstrated using the weak measurement method. The polarization rotation angle
measurement sensitivity (\delta \psi ) of 16 \mu~rad and hence of the
magnetometer of 1 nT, achieved using the weak measurement method is better than
the balanced optical polarimetry results by a factor of three. The improvement
in the measurement sensitivity is realized via optical amplification of the
polarization rotation angle via spin-orbit coupled light beam-field. The method
is devoid of external rf modulation, allows for optimal tunability of
sensitivity depending on the dynamic range of the applied magnetic field and
the sensitivity can be further enhanced by operating in the Spin Exchange
Relaxation Free regime of alkali spin polarization.
