  We present the highest resolution --- 15 pc (0.03'') --- ALMA $^{12}$CO(2-1)
line emission and 1.3mm continuum maps, tracers of the molecular gas and dust,
respectively, in the nearby merging galaxy system NGC 6240, that hosts two
supermassive black holes growing simultaneously. These observations provide an
excellent spatial match to existing Hubble optical and near-infrared
observations of this system. A significant molecular gas mass,
$\sim$9$\times$10$^9$M$_\odot$, is located in between the two nuclei, forming a
clumpy stream kinematically dominated by turbulence, rather than a smooth
rotating disk as previously assumed from lower resolution data. Evidence for
rotation is seen in the gas surrounding the southern nucleus, but not in the
northern one. Dynamical shells can be seen, likely associated with nuclear
supernovae remnants. We further detect the presence of significant high
velocity outflows, some of them reaching velocities $>$500 km/s, affecting a
significant fraction, $\sim$11\% of the molecular gas in the nuclear region.
Inside the spheres of influence of the northern and southern supermassive black
holes we find molecular masses of 7.4$\times$10$^8$M$_\odot$ and
3.3$\times$10$^9$M$_\odot$, respectively. We are thus directly imaging the
reservoir of gas that can accrete onto each supermassive black hole. These new
ALMA maps highlight the critical need for high resolution observations of
molecular gas in order to understand the feeding of supermassive black holes
and its connection to galaxy evolution in the context of a major galaxy merger.
