  We demonstrate a new method for measuring the escape fraction of ionizing
photons using Hubble Space Telescope imaging of resolved stars in NGC 4214, a
local analog of high-redshift starburst galaxies that are thought to be
responsible for cosmic reionization. Specifically, we forward model the UV
through near-IR spectral energy distributions of $\sim$83,000 resolved stars to
infer their individual ionizing flux outputs. We constrain the local escape
fraction by comparing the number of ionizing photons produced by stars to the
number that are either absorbed by dust or consumed by ionizing the surrounding
neutral hydrogen in individual star-forming regions. We find substantial
spatial variation in the escape fraction (0-40%). Integrating over the entire
galaxy yields a global escape fraction of 25% (+16%/-15%). This value is much
higher than previous escape fractions of zero reported for this galaxy. We
discuss sources of this apparent tension, and demonstrate that the viewing
angle and the 3D ISM geometric effects are the cause. If we assume the NGC 4214
has no internal dust, like many high-redshift galaxies, we find an escape
fraction of 59% (an upper-limit for NGC 4214). This is the first non-zero
escape fraction measurement for UV-faint (M$_{\rm FUV}$ = -15.9) galaxies at
any redshift, and supports the idea that starburst UV-faint dwarf galaxies can
provide a sufficient amount of ionizing photons to the intergalactic medium.
