  We present moderate (${\sim}5^{\prime\prime}$) and high angular resolution
(${\sim}1^{\prime\prime}$) observations of $^{12}\rm{CO\,}(J=2-1)$ emission
toward nearby, interacting galaxy NGC 3627 taken with the Submillimeter Array
(SMA). These SMA mosaic maps of NGC 3627 reveal a prominent nuclear peak,
inter-arm regions, and diffuse, extended emission in the spiral arms. A
velocity gradient of ${\sim}400$-$450$ km s$^{-1}$ is seen across the entire
galaxy with velocity dispersions ranging from $\lesssim 80$ km s$^{-1}$ toward
the nuclear region to $\lesssim 15$ km s$^{-1}$ in the spiral arms. We also
detect unresolved $^{13}\rm{CO\,}(J=2-1)$ line emission toward the nuclear
region, southern bar end, and in a relatively isolated clump in the southern
portion of the galaxy, while no $\rm{C}^{18}O(J=2-1)$ line emission is detected
at a $3\sigma$ rms noise level of 42 mJy beam$^{-1}$ per 20 km s$^{-1}$
channel. Using RADEX modeling with a large velocity gradient approximation, we
derive kinetic temperatures ranging from ${\sim}5$-$10$ K (in the spiral arms)
to ${\sim}25$ K (at the center) and H$_2$ number densities from
${\sim}$400-1000 cm$^{-3}$ (in the spiral arms) to ${\sim}$12500 cm$^{-3}$ (at
the center). From this density modeling, we find a total H$_2$ mass of
$9.6\times10^9 M_{\odot}$, which is ${\sim}50\%$ higher than previous estimates
made using a constant H$_2$-CO conversion factor but is largely dependent on
the assumed vertical distribution of the CO gas. With the exception of the
nuclear region, we also identify a tentative correlation between star formation
efficiency and kinetic temperature. We derive a galactic rotation curve,
finding a peak velocity of ${\sim}207$ km s$^{-1}$ and estimate a total
dynamical mass of $4.94 \pm 0.70 \times 10^{10} M_{\odot}$ at a galactocentric
radius of ${\sim}6.2$ kpc ($121^{\prime\prime}$).
