Galaxies observed at redshift z > 6, when the Universe was less than a billion years old, thus far very rarely show evidence of the cold dust that accompanies star formation in the local Universe, where the dust-to-gas mass ratio is around one per cent. A prototypical example is the galaxy Himiko (z = 6.6), which—a mere 840 million years after the Big Bang—is forming stars at a rate of 30–100 solar masses per year, yielding a mass assembly time of about 150 × 106 years. Himiko is thought to have a low fraction (2–3 per cent of the Sun’s) of elements heavier than helium (low metallicity), and although its gas mass cannot yet be determined its dust-to-stellar mass ratio is constrained to be less than 0.05 per cent. The local dwarf galaxy I Zwicky 18, which has a metallicity about 4 per cent that of the Sun’s and is forming stars less rapidly (assembly time about 1.6 × 109 years) than Himiko but still vigorously for its mass, is also very dust deficient and is perhaps one of the best analogues of primitive galaxies accessible to detailed study. Here we report observations of dust emission from I Zw 18, from which we determine its dust mass to be 450–1,800 solar masses, yielding a dust-to-stellar mass ratio of about 10−6 to 10−5 and a dust-to-gas mass ratio of 3.2–13 × 10−6. If I Zw 18 is a reasonable analogue of Himiko, then Himiko’s dust mass must be around 50,000 solar masses, a factor of 100 below the current upper limit. These numbers are quite uncertain, but if most high-z galaxies are more like Himiko than like the very-high-dust-mass galaxy SDSS J114816.64 + 525150.3 at z ≈ 6, which hosts a quasar, then our prospects for detecting the gas and dust inside such galaxies are much poorer than hitherto anticipated.