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The small grain sizes produced by Type II supernova models in young, metal-poor galaxies make the appearance of their infrared (IR) spectral energy distribution (SED) quite different from that of nearby, older galaxies. To study this effect, we have developed a model for the evolution of dust content and the IR SED of low-metallicity, extremely young galaxies based on recent work by Hirashita et al. We find that, even in the intense ultraviolet radiation field of very young galaxies, small silicate grains are subject to stochastic heating resulting in a broad temperature distribution and substantial mid-infrared (MIR) continuum emission. Larger carbonaceous grains are in thermal equilibrium at T ≃ 50-100 K, and they also contribute to the MIR. We present the evolution of SEDs and IR extinction of very young, low-metallicity galaxies. The IR extinction curve is also shown. In the first few Myr, the emission peaks at λ∼ 30-50 µm; at later times, dust self-absorption decreases the apparent grain temperatures, shifting the bulk of the emission into the submillimetre band. We successfully apply the model to the IR SED of SBS 0335-052, a low-metallicity (1/41 Z_☉) dwarf galaxy with an unusually strong MIR flux. We find the SED, optical properties and extinction of the star-forming region to be consistent with a very young (age ≃ 6.5x10⁶ yr) and compact (radius ≃ 20 pc) starburst. We also predict the SED of another extremely low-metallicity galaxy, IZw 18, for future observational tests. We estimate the FIR luminosity of IZw 18 to be low as L_FIR∼ 10⁷-10^7.5 L_☉, depending on the uncertainty of dust mass. Some prospects for future observations are discussed.