Recovery of the supernova (SN) delay-time distribution (DTD) - the SN rate versus time that would follow a hypothetical brief burst of star formation - can shed light on SN progenitors and physics, as well as on the time-scales of chemical enrichment. Previous attempts to reconstruct the DTD have been based either on comparison of mean SN rates versus redshift to cosmic star-formation history (SFH), or on the comparison of SN rates among galaxies with different mean ages. Here, we present an approach to recover the SN DTD that avoids the averaging and loss of information of other schemes. We compare the SFHs of individual galaxies to the numbers of SNe discovered by a survey in each galaxy (generally zero, sometimes one SN, rarely a few). We apply the method to a subsample of 3505 galaxies, hosting 82 type-Ia SNe (SNe Ia) and 119 core-collapse supernovae (CC SNe), from the Lick Observatory Supernova Search (LOSS), that have SFHs reconstructed from Sloan Digital Sky Survey (SDSS) spectra. We find a >2σ SN Ia DTD signal in our shortest-delay, 'prompt' bin at <420 Myr. We identify and study a systematic error, due to the limited aperture of the SDSS spectroscopic fibres, that causes some of the prompt signal to leak to the later bins of the DTD. After accounting for this systematic error, we demonstrate that a prompt SN Ia contribution is required by the data at the >99 per cent confidence level. We further find a 4σ indication of SNe Ia that are 'delayed' by >2.4 Gyr. Thus, the data support the existence of both prompt and delayed SNe Ia. We measure the time integral over the SN DTD. For CC SNe we find a total yield of 0.010 ± 0.002 SNe per M⊙ formed, in excellent agreement with expectations, if all stars more massive than 8 M⊙ lead to visible SN explosions. This argues against scenarios in which the minimum mass for core-collapse SNe is ≳10 M⊙, or in which a significant fraction of massive stars collapse without an accompanying explosion. For SNe Ia, the time-integrated yield is 0.0023 ± 0.0006 SNe per M⊙ formed, most of them with delays <2.4 Gyr. Finally, we show the robust performance of the method on simulated samples, and demonstrate that its application to already existing SN samples, such as the full LOSS sample, but with complete and unbiased SFH estimates for the survey galaxies, could provide an accurate and detailed measurement of the SN Ia DTD.