We report a Plateau de Bure interferometer search for CII-158$mu$m emission from HCM6A, a lensed Lyman-$alpha$ emitter (LAE) at $z = 6.56$. Our non-detections of CII-158$mu$m line emission and 1.2mm radio continuum emission yield $3sigma$ limits of L$_{rm CII} < 6.4 times 10^7 times (Delta V/100 km s^{-1})^{1/2}$ L$_odot$ on the CII-158$mu$m line luminosity and S$_{rm 1.2mm} < 0.68$ mJy on the 1.2mm flux density. The local conversion factor between L$_{rm CII}$ and star formation rate (SFR) yields an SFR $< 4.7$ M$_odot$ yr$^{-1}$, $approx 2$ times lower than that inferred from the ultraviolet (UV) continuum, suggesting that the local factor may not be applicable in high-$z$ LAEs. The non-detection of 1.2mm continuum emission yields a total SFR $< 28$ M$_odot$/yr; any obscured star formation is thus within a factor of two of the visible star formation. Our best-fit model to the rest-frame UV/optical spectral energy distribution of HCM6A yields a stellar mass of $1.3 times 10^9$ M$_odot$ and an SFR of ~10 M$_odot$/yr, with negligible dust obscuration. We fortuitously detect CO J=3-2 emission from a $z=0.375$ galaxy in the foreground cluster Abell370, obtaining a CO line luminosity of L$^prime ({rm CO}) > (8.95 pm 0.79) times 10^8$ K km s$^{-1}$ pc$^2$, and a molecular gas mass of M$({rm H_2}) > (4.12 pm 0.36) times 10^9$ M$_odot$, for a CO-to-H$_2$ conversion factor of 4.6 M$_odot$ (K km s$^{-1}$ pc$^2$)$^{-1}$.
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