Top-quark pair production is observed in lead-lead (Pb+Pb) collisions at sqrt[s_{NN}]=5.02  TeV at the Large Hadron Collider with the ATLAS detector. The data sample was recorded in 2015 and 2018, amounting to an integrated luminosity of 1.9  nb^{-1}. Events with exactly one electron and one muon and at least two jets are selected. Top-quark pair production is measured with an observed (expected) significance of 5.0 (4.1) standard deviations. The measured top-quark pair production cross section is σ_{tt[over ¯]}=3.6 _{-0.9}^{+1.0}(stat) _{-0.5}^{+0.8}(syst)  μb, with a total relative uncertainty of 31%, and is consistent with theoretical predictions using a range of different nuclear parton distribution functions. The observation of this process consolidates the evidence of the existence of all quark flavors in the preequilibrium stage of the quark-gluon plasma at very high energy densities, similar to the conditions present in the early Universe.