[en] We consider heat transport in one-dimensional harmonic chains with isotopic disorder, focusing our attention mainly on how disorder correlations affect heat conduction. Our approach reveals that long-range correlations can change the number of low-frequency extended states. As a result, with a proper choice of correlations one can control how the conductivity κ scales with the chain length N. We present a detailed analysis of the role of specific long-range correlations for which a size-independent conductivity is exactly recovered in the case of fixed boundary conditions. As for free boundary conditions, we show that disorder correlations can lead to a conductivity scaling as $\mathrm{\kappa <!--\; ??\; -->}\sim <!--\; ???\; -->N^{\mathrm{\epsilon <!--\; ??\; -->}}$, with the scaling exponent ε being arbitrarily small (although not strictly zero), so that normal conduction is almost recovered even in this case. (letter)