The Type Ia supernova (SN Ia) rate, when compared to the cosmic star formation history (SFH), can be used to derive the delay-time distribution (DTD; the hypothetical SN Ia rate versus time following a brief burst of star formation) of SNe Ia, which can distinguish among progenitor models. We present the results of a supernova (SN) survey in the Subaru Deep Field (SDF). Over a period of 3 years, we have observed the SDF on four independent epochs with Suprime-Cam on the Subaru 8.2-m telescope, with two nights of exposure per epoch, in the R, i′and z′ bands. We have discovered 150 SNe out to redshift z≈ 2. Using 11 photometric bands from the observer-frame far-ultraviolet to the near-infrared, we derive photometric redshifts for the SN host galaxies (for 24 we also have spectroscopic redshifts). This information is combined with the SN photometry to determine the type and redshift distribution of the SN sample. Our final sample includes 28 SNe Ia in the range 1.0 < z < 1.5 and 10 in the range 1.5 < z < 2.0. As our survey is largely insensitive to core-collapse SNe (CC SNe) at z > 1, most of the events found in this range are likely SNe Ia. Our SN Ia rate measurements are consistent with those derived from the Hubble Space Telescope (HST) Great Observatories Origins Deep Survey (GOODS) sample, but the overall uncertainty of our 1.5 < z < 2.0 measurement is a factor of 2 smaller, of 35-50 per cent. Based on this sample, we find that the SN Ia rate evolution levels off at 1.0 < z < 2.0, but shows no sign of declining. Combining our SN Ia rate measurements and those from the literature, and comparing to a wide range of possible SFHs, the best-fitting DTD (with a reduced χ2= 0.7) is a power law of the form Ψ(t) ∝t β, with index β=−1.1 ± 0.1 (statistical) ±0.17 (systematic). This result is consistent with other recent DTD measurements at various redshifts and environments, and is in agreement with a generic prediction of the double-degenerate progenitor scenario for SNe Ia. Most single-degenerate models predict different DTDs. By combining the contribution from CC SNe, based on the wide range of SFHs, with that from SNe Ia, calculated with the best-fitting DTD, we predict that the mean present-day cosmic iron abundance is in the range Z Fe= (0.09-0.37) ZFe, ⊙. We further predict that the high-z SN searches now beginning with HST will discover 2-11 SNe Ia at z > 2.