Pricing with volatility that varies over time

Suppose a stock's volatility is not constant but a known deterministic function $\sigma(t)$. How do you price and hedge a European option on it?

Use Black-Scholes with the root-mean-square volatility over the remaining life. Log-returns are still normal, with variance $\int_t^T\sigma(s)^2ds$, so the price is the Black-Scholes formula with $\bar\sigma^2=\tfrac{1}{T-t}\int_t^T\sigma(s)^2ds$. Hedge with the corresponding delta, recomputed at each date with the variance still to come.