We study the formation of molecular precursors to dust in the ejecta of Population III supernovae using a chemical kinetic approach to follow the evolution of small dust cluster abundances from day 100 to day 1000 after explosion. Our work focuses on zero-metallicity 20 M ⊙ and 170 M ⊙ progenitors, and we consider fully-macroscopically mixed and unmixed ejecta. The dust precursors comprise molecular chains, rings and small clusters of chemical composition relevant to the initial elemental composition of the ejecta under study. The nucleation stage for small silica, metal oxides and sulphides, pure metal, and carbon clusters is described with a new chemical reaction network highly relevant to the kinetic description of dust formation in hot circumstellar environments. We consider the effect of the pressure dependence of critical nucleation rates, and test the impact of microscopically-mixed He + on carbon dust formation. Two cases of metal depletion on silica clusters (full and no depletion) are considered to derive upper limits to the amounts of dust produced in SN ejecta at 1000 days, while the chemical composition of clusters gives a prescription for the type of dust formed in Pop. III supernovae.We show that the cluster mass produced in the fully-mixed ejecta of a 170M ⊙ progenitor is ∼ 25 M ⊙ whereas its 20 M ⊙ counterpart forms ∼ 0.16 M ⊙ of clusters. The unmixed ejecta of a 170 M ⊙ progenitor supernova synthesizes ∼ 5.6 M ⊙ of small clusters, while its 20 M ⊙ counterpart produces ∼ 0.103 M ⊙ . Our results point to smaller amounts of dust formed in the ejecta of Pop. III supernovae by a factor ∼ 5 compared to values derived by previous studies, and to different dust chemical composition. Such deviations result from some erroneous assumptions made, the inappropriate use of classical nucleation theory to model dust formation, and the omission of the synthethis of molecules in supernova ejecta. We also find that the unmixed ejecta of massive Pop. III supernovae chiefly form silica and/or silicates, and pure silicon grains whereas their lower mass counterparts form a dust mixture dominated by silica and/or silicates, pure silicon and iron sulphides. Amorphous carbon can only condense in ejecta where the carbon-rich zone is deprived of He + via the nucleation of carbon chains and rings characteristic of the synthesis of fullerenes. The first dust enrichment to the primordial gas in the early universe from Pop. III massive supernova comprises primarily pure silicon, silica and silicates. If carbon dust is present at redshift z > 6, alternative dust sources must be considered.