  We observe a series of conical intersections in the potential energy curves
governing both the collision between a Rydberg atom and a ground-state atom and
the structure of Rydberg molecules. By employing the electronic energy of the
Rydberg atom as a synthetic dimension we circumvent the von Neumann-Wigner
theorem. These conical intersections can occur when the Rydberg atom's quantum
defect is similar in size to the electron--ground-state atom scattering phase
shift divided by $\pi$, a condition satisfied in several commonly studied
atomic species. The conical intersections have an observable consequence in the
rate of ultracold $l$-changing collisions of the type Rb$(nf)$+Rb$(5s)\to$
Rb$(nl>3)$+Rb$(5s)$. In the vicinity of a conical intersection, this rate is
strongly suppressed, and the Rydberg atom becomes nearly transparent to the
ground-state atom.
