We analyze the temperature anomaly of the Pinatubo eruption using an exact mathematical solution of a standard energy balance model that includes coupling between the mixed layer and the thermocline. Our solution yields a short response time τ = 4.4 months and a small climate sensitivity λ = 0.22 C/(W/m2), implying short-term negative feedback. Also, our analysis determines a value of the effective eddy diffusion constant $kappa$ = 2 ¿ 10-6 m2/s that is much smaller than that assumed in many climate models. We find for this model that the heat flux to the thermocline reverses sign and integrates to zero for any forcing of finite duration. This effect should be observable in any future Pinatubo-type event.