We present Hubble Space Telescope UV spectra of the 4.6 h period double white dwarf SDSS J125733.63+542850.5. Combined with Sloan Digital Sky Survey optical data, these reveal that the massive white dwarf (secondary) has an effective temperature T2 = 13030 +/- 70 +/- 150 K and a surface gravity log g2 = 8.73 +/- 0.05 +/- 0.05 (statistical and systematic uncertainties respectively), leading to a mass of M2 = 1.06 Msun. The temperature of the extremely low-mass white dwarf (primary) is substantially lower at T1 = 6400 +/- 37 +/- 50 K, while its surface gravity is poorly constrained by the data. The relative flux contribution of the two white dwarfs across the spectrum provides a radius ratio of R1/R2 = 4.2, which, together with evolutionary models, allows us to calculate the cooling ages. The secondary massive white dwarf has a cooling age of about 1 Gyr, while that of the primary low-mass white dwarf is likely to be much longer, possibly larger than 5 Gyrs, depending on its mass and the strength of chemical diffusion. These results unexpectedly suggest that the low-mass white dwarf formed long before the massive white dwarf, a puzzling discovery which poses a paradox for binary evolution. ; Comment: accepted for publication in MNRAS