Given $\nu>1/2$ and $\delta>0$ arbitrary, we prove the existence of energy solutions of \begin{equation}\partial_{tt} u - \Delta u - u^5 =0~~~~(0.1)\end{equation} in ${\mathbb R}^{3+1}$ which blow up exactly at $r=t=0$ as $t \to 0-$ . These solutions are radial and of the form $u = \lambda(t)^{1/2} W(\lambda(t)r) + \eta(r,t)$ inside the cone $r\le t$ , where $\lambda(t)=t^{-1-\nu}$ , $W(r)=(1+r^2/3)^{-1/2}$ is the stationary solution of (0.1), and $\eta$ is a radiation term with \[ \int_{[r\le t]} \big(|\nabla \eta(x,t)|^2 + |\eta_t(x,t)|^2+|\eta(x,t)|^6\big) dx \to 0, \quad t \to 0. \] Outside of the light-cone, there is the energy bound \[ \int_{[r>t]} \big( |\nabla u(x,t)|^2+|u_t(x,t)|^2+|u(x,t)|^6\big) dx \lt \delta \] \] for all small $t>0$ . The regularity of $u$ increases with $\nu$ . As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” $W(r)$