  Developing robust microwave-field sensors is both fundamentally and
practically important with a wide range of applications from astronomy to
communication engineering. The Nitrogen-Vacancy (NV) center in diamond is an
attractive candidate for such purpose because of its magnetometric sensitivity,
stability and compatibility with ambient conditions. However, the existing NV
center-based magnetometers have limited sensitivity in the microwave band. Here
we present a continuous heterodyne detection method that can enhance the
sensor's response to weak microwaves, even in the absence of spin controls.
Experimentally, we achieve a sensitivity of 8.9 pT$\cdot$Hz$^{-1/2}$ for
microwaves of 2.9 GHz by simultaneously using an ensemble of $n_{\text{NV}}
\sim 2.8\times10^{13}$ NV centers within a sensor volume of $4\times10^{-2}$
mm$^3$. Besides, we also achieve $1/t$ scaling of frequency resolution up to
measurement time $t$ of 10000 s. Our method removes the control pulses and thus
will greatly benefit the practical application of diamond-based microwave
sensors.
