We study optimal climate policy when the low-carbon transition is constrained by frictions in scaling clean energy and retiring fossil assets. We develop a stochastic, quantitative, intertemporal general-equilibrium model with heterogeneous dirty capital disciplined using firm-level data and adjustment costs disciplined using evidence on historical energy-system expansions and phase-outs. The data reveal an important `dirty tail‘ of highly emissions-intensive assets that can be retired early at low cost. This means that, despite transition frictions, optimal policy entails rapid near-term decarbonisation, with emissions halving in the first ten years. The transition is not primarily constrained by the cost of stranding dirty capital, but by the rate at which clean capital can be expanded. Our results highlight the importance of modelling heterogeneity in emissions intensity and support policies targeting high-emissions assets and easing clean investment constraints.