Current observations favour that the massive ultraviolet-bright clumps with a median stellar mass of $\sim 10^7\, {\rm M}_{\odot }$, ubiquitously observed in z ∼ 1–3 galaxies, are star-forming regions formed in situ in galaxies. It has been proposed that they result from gas fragmentation due to gravitational instability of gas-rich, turbulent, and high-redshift discs. We bring support to this scenario by reporting the new discovery of giant molecular clouds (GMCs) in the strongly lensed, clumpy, main-sequence galaxy, A521-sys1, at z = 1.043. Its CO(4–3) emission was mapped with the Atacama Large Millimetre/submillimetre Array (ALMA) at an angular resolution of 0.19 × 0.16 arcsec2, reading down to 30 pc, thanks to gravitational lensing. We identified 14 GMCs, most being virialized, with $10^{5.9}-10^{7.9}\, {\rm M}_{\odot }$ masses and a median $800\, {\rm M}_{\odot }~\mathrm{pc}^{-2}$ molecular gas mass surface density, that are, respectively, 100 and 10 times higher than for nearby GMCs. They are also characterized by 10 times higher supersonic turbulence with a median Mach number of 60. They end up to fall above the Larson scaling relations, similarly to the GMCs in another clumpy z ≃ 1 galaxy, the Cosmic Snake, although differences between the two sets of high-redshift GMCs exist. Altogether they support that GMCs form with properties that adjust to the ambient interstellar medium conditions prevalent in the host galaxy whatever its redshift. The detected A521-sys1 GMCs are massive enough to be the parent gas clouds of stellar clumps, with a relatively high star formation efficiency per free-fall time of ∼11 per cent.