  Primordial magnetic fields generated in the early universe are subject of
considerable investigation, and observational limits on their strength are
required to constrain the theory. Due to their impact on the reionization
process, the strength of primordial fields can be limited using the latest data
on reionization and the observed UV-luminosity function of high-redshift
galaxies. Given the steep faint-end slope of the luminosity function, faint
galaxies contribute substantial ionizing photons, and the low-luminosity cutoff
has an impact on the total budget thereof. Magnetic pressure from primordial
fields affects such cutoff by preventing collapse in halos with mass below
10^{10} M_solar (B_0 / 3 nG)^3, with B_0 the co-moving field strength. In this
letter, the implications of these effects are consistently incorporated in a
simplified model for reionization, and the uncertainties due to the
cosmological parameters, the reionization parameters and the observed UV
luminosity function are addressed. We show that the observed ionization degree
at z\sim7 leads to the strongest upper limit of B_0\lsim 2-3nG. Stronger limits
could follow from measurements of high ionization degree at z>7.
