One of the most intriguing features of some high-temperature cuprate superconductors is the interplay between one-dimensional “striped” spin order and charge order, and superconductivity. We used mid-infrared femtosecond pulses to transform one such stripe-ordered compound, nonsuperconducting La1.675Eu0.2Sr0.125CuO4, into a transient three-dimensional superconductor. The emergence of coherent interlayer transport was evidenced by the prompt appearance of a Josephson plasma resonance in the c-axis optical properties. An upper limit for the time scale needed to form the superconducting phase is estimated to be 1 to 2 picoseconds, which is significantly faster than expected. This places stringent new constraints on our understanding of stripe order and its relation to superconductivity.