We study an inhomogeneous branching Brownian motion in which individual particles execute standard Brownian movements and reproduce at rates depending on their locations. The rate of reproduction for a particle located at $x$ is $\beta(x) = b + \beta_0(x)$, where $\beta_0(x)$ is a nonnegative, continuous, integrable function. Let $M(t)$ be the position of the rightmost particle at time $t$; then as $t \rightarrow \infty, M(t) - \operatorname{med}(M(t))$ converges in law to a location mixture of extreme value distributions. We determine $\operatorname{med}(M(t))$ to within a constant $+ o(1)$. The rate at which $\operatorname{med}(M(t)) \rightarrow \infty$ depends on the largest eigenvalue $\lambda$ of a differential operator involving $\beta(x)$; the cases $\lambda < 2, \lambda = 2$ and $\lambda > 2$ are qualitatively different.