  One of the most striking universal properties of the
high-transition-temperature (high-$T_c$) superconductors is that they are all
derived from the hole-doping of their insulating antiferromagnetic (AF) parent
compounds. From the outset, the intimate relationship between magnetism and
superconductivity in these copper-oxides has intrigued researchers. Evidence
for this link comes from neutron scattering experiments that show the
unambiguous presence of short-range AF correlations (excitations) in cuprate
superconductors. Even so, the role of such excitations in the pairing mechanism
and superconductivity is still a subject of controversy. For
YBa$_2$Cu$_3$O$_{6+x}$, where $x$ controls the hole-doping level, the most
prominent feature in the magnetic excitations spectra is the ``resonance''.
Here we show that for underdoped YBa$_2$Cu$_3$O$_{6.6}$, where $x$ and $T_c$
are below the optimal values, modest magnetic fields suppress the resonance
significantly, much more so for fields approximately perpendicular rather than
parallel to the CuO$_2$ planes. Our results indicate that the resonance
measures pairing and phase coherence, suggesting that magnetism plays an
important role in the superconductivity of cuprates. The persistence of a field
effect above $T_c$ favors mechanisms with preformed pairs in the normal state
of underdoped cuprates.
