The possibility that population III stars have reionized the Universe at redshifts greater than 6 has recently gained momentum with WMAP polarization results. Here we analyse the role of early dust produced by these stars and ejected into the intergalactic medium. We show that this dust, heated by the radiation from the same population III stars, produces a submillimetre excess. The electromagnetic spectrum of this excess could account for a significant fraction of the FIRAS (Far Infrared Absolute Spectrophotometer) cosmic far infrared background above 700 micron. This spectrum, a primary anisotropy (Delta T) spectrum times the nu2 dust emissivity law, peaking in the submillimetre domain around 750 micron, is generic and does not depend on other detailed dust properties. Arcminute-scale anisotropies, coming from inhomogeneities in this early dust, could be detected by future submillimetre experiments such as Planck HFI.