We consider the parameter space ${\cal U}_d$ of smooth plane curves of degree $d$. The universal smooth plane curve of degree $d$ is a fiber bundle ${\cal E}_d\to{\cal U}_d$ with fiber diffeomorphic to a surface $\Sigma_g$. This bundle gives rise to a monodromy homomorphism $\rho_d:\pi_1({\cal U}_d)\to{\rm Mod}(\Sigma_g)$, where ${\rm Mod}(\Sigma_g):=\pi_0({\rm Diff}^+(\Sigma_g))$ is the mapping class group of $\Sigma_g$. The main result of this paper is that the kernel of $\rho_4:\pi_1({\cal U}_4)\to{\rm Mod}(\Sigma_3)$ is isomorphic to $F_\infty\times{\bb Z}/3{\bb Z}$, where $F_\infty$ is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement ${\rm Teich}(\Sigma_g)\setminus{\cal H}_g$ of the hyperelliptic locus ${\cal H}_g$ in Teichm\"uller space ${\rm Teich}(\Sigma_g)$ has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil-Petersson geometry of Teichm\"uller space together with results from algebraic geometry.