  We present the first observations of H$^{13}$CN$(1-0)$, H$^{13}$CO$^+(1-0)$
and SiO$(2-1)$ in NGC\,6240, obtained with the IRAM PdBI. Combining a Markov
Chain Monte Carlo (MCMC) code with Large Velocity Gradient (LVG) modelling, and
with additional data from the literature, we simultaneously fit three gas
phases and six molecular species to constrain the physical condition of the
molecular gas, including mass$-$luminosity conversion factors. We find
$\sim10^{10}M_\odot$ of dense molecular gas in cold, dense clouds ($T_{\rm
k}\sim10$\,K, $n_{{\rm H}_2}\sim10^6$\,cm$^{-3}$) with a volume filling factor
$<0.002$, embedded in a shock heated molecular medium ($T_{\rm k}\sim2000$\,K,
$n_{{\rm H}_2}\sim10^{3.6}$\,cm$^{-3}$), both surrounded by an extended diffuse
phase ($T_{\rm k}\sim200$\,K, $n_{{\rm H}_2}\sim10^{2.5}$\,cm$^{-3}$). We
derive a global $\alpha_{\rm CO}=1.5^{7.1}_{1.1}$ with gas masses
$\log_{10}\left(M / [M_\odot]\right)=10.1_{10.0}^{10.8}$, dominated by the
dense gas. We also find $\alpha_{\rm HCN} = 32^{89}_{13}$, which traces the
cold, dense gas. The [$^{12}$C]/[$^{13}$C] ratio is only slightly elevated
($98^{230}_{65}$), contrary to the very high [CO]/[$^{13}$CO] ratio (300-500)
reported in the literature. However, we find very high [HCN]/[H$^{13}$CN] and
[HCO$^+$]/[H$^{13}$CO$^+$] abundance ratios $(300^{500}_{200})$ which we
attribute to isotope fractionation in the cold, dense clouds.
