The chemistry of planet-forming disks sets the exoplanet atmosphere composition and the prebiotic molecular content. Dust traps are of particular importance as pebble growth and transport are crucial for setting the chemistry where giant planets are forming. The asymmetric Oph~IRS~48 dust trap located at 60 au radius provides a unique laboratory for studying chemistry in pebble-concentrated environments in warm Herbig disks with low gas-to-dust ratios down to 0.01. We use deep ALMA Band~7 line observations to search the IRS~48 disk for H$_2$CO and CH$_3$OH line emission, the first steps of complex organic chemistry. We report the detection of 7 H$_2$CO and 6 CH$_3$OH lines with energy levels between 17 and 260 K. The line emission shows a crescent morphology, similar to the dust continuum, suggesting that the icy pebbles play an important role in the delivery of these molecules. Rotational diagrams and line ratios indicate that both molecules originate from warm molecular regions in the disk with temperatures $>$100 K and column densities $sim10^{14}$ cm$^{-2}$ or a fractional abundance of $sim10^{-8}$ and with H$_2$CO/CH$_3$OH$sim$0.2, indicative of ice chemistry. Based on arguments from a physical-chemical model with low gas-to-dust ratios, we propose a scenario where the dust trap provides a huge icy grain reservoir in the disk midplane or an `ice trap, which can result in high gas-phase abundances of warm COMs through efficient vertical mixing. This is the first time that complex organic molecules have been clearly linked to the presence of a dust trap. These results demonstrate the importance of including dust evolution and vertical transport in chemical disk models, as icy dust concentrations provide important reservoirs for complex organic chemistry in disks.
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