We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $α<1/7$, both of them are $α$-Hölder continuous and $(\varrho, u)$ is a weak solution to the underlying equations. The proof is based on typical convex integration techniques using Mikado flows as building blocks. As a main novelty with respect to the related literature, our result produces a Hölder continuous density.