  We present a magnetic sensor with energy resolution per bandwidth $E_R <
\hbar$. We show how a $^{87}\mathrm{Rb}$ single domain spinor Bose-Einstein
condensate, detected by non-destructive Faraday-rotation probing, achieves
single shot dc magnetic sensitivity of $72(8)~\mathrm{fT}$ measuring a volume
$V= 1091(30)~\mu\mathrm{m}^3$ for $3.5~\mathrm{s}$, and thus $E_R =
0.075(16)~\hbar$. We measure experimentally the condensate volume, spin
coherence time, and readout noise, and use phase-space methods, backed by 3+1D
mean-field simulations, to compute the spin noise. Contributions to the spin
noise include one-body and three-body losses and shearing of the projection
noise distribution, due to competition of ferromagnetic contact interactions
and quadratic Zeeman shifts. Nonetheless, the fully-coherent nature of the
single-domain, ultracold two-body interactions allows the system to escape the
coherence vs.~density trade-off that imposes an energy resolution limit on
traditional spin-precession sensors. We predict that other Bose-condensed
alkalis, especially the antiferromagnetic $^{23}\mathrm{Na}$, can further
improve the energy resolution of this method.
