  We solve the longstanding conjecture by Milnor (1993) concerning the
connectedness locus $M_1$ of the family of quadratic rational maps tangent to
the identity at $\infty$. We prove that this locus in homeomorphic to the
Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies
maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this
homeomorphism from $M$ to $M_1$ is nowhere H\"older on the boundary and so can
not have even locally a quasi-conformal extension to complements.
