It is widely accepted that within the framework of Λ cold dark matter a significant fraction of giant-disc galaxies has recently experienced a violent galactic merger. We present numerical simulations of such major mergers of gas-rich pure disc galaxies, and focus on the innermost stellar component (bulge) of the disc remnants. The simulations have high spatial and mass resolutions, and resolve regions deep enough to allow bulge classification according to standard kinematical and structural characteristics. In agreement with recent studies we find that these bulges are dominated by stars formed in the final coalescence process. In contrast to the common interpretation of such components as classical bulges (i.e. similar to intermediate-luminosity ellipticals), we find that they are supported by highly coherent rotations and have Sèrsic indices n < 2, a result leading to their classification as pseudo-bulges. Pseudo-bulge formation by gas-rich major mergers of pure discs is a novel mode of pseudo-bulge formation; it complements pseudo-bulge growth by secular evolution, and it could help explain the high fractions of classically bulgeless giant-disc galaxies, and pseudo-bulges found in giant Sc galaxies.