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New Early Dark Energy (NEDE) is a component of vacuum energy at the electron volt scale, which decays in a first-order phase transition shortly before recombination [arXiv:1910.10739]. The NEDE component has the potential to resolve the tension between recent local measurements of the expansion rate of the Universe using supernovae (SN) data and the expansion rate inferred from the early Universe through measurements of the cosmic microwave background (CMB) when assuming $Λ$CDM. We discuss in depth the two-scalar field model of the NEDE phase transition including the process of bubble percolation, collision, and coalescence. We also estimate the gravitational wave signal produced during the collision phase and argue that it can be searched for using pulsar timing arrays. In a second step, we construct an effective cosmological model, which describes the phase transition as an instantaneous process, and derive the covariant equations that match perturbations across the transition surface. Fitting the cosmological model to CMB, baryonic acoustic oscillations and SN data, we report $H_0 = 69.6^{+1.0}_{-1.3} \, \textrm{km}\, \textrm{s}^{-1}\, \textrm{Mpc}^{-1}$ $(68 \%$ C.L.) without the local measurement of the Hubble parameter, bringing the tension down to $2.5\, σ$. Including the local input, we find $H_0 = 71.4 \pm 1.0 \, \textrm{km}\, \textrm{s}^{-1}\, \textrm{Mpc}^{-1}$ $(68 \%$ C.L.) and strong evidence for a non-vanishing NEDE component with a $\simeq 4\, σ$ significance.