We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-N\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitean $N \times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. We prove that the two-point function admits an explicit solution formula in terms of the roots of a meromorphic function $J$ constructed from the spectrum of $E$. Structures which appear in this solution can be assembled into complex curves. We also solve the large-$N$ limit to unbounded operators $E$. The renormalised two-point function is given by an integral formula involving a regularisation of $J$. We prove triviality of the renormalised four-dimensional quartic matrix model for all positive $\lambda$.