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The production of the hypertriton nuclei 3ΛH and 3Λ¯H¯¯¯¯ has been measured for the first time in Pb-Pb collisions at sNN−−−√ = 2.76 TeV with the ALICE experiment at LHC energies. The total yield, dN/dy ×B.R.(3ΛH→3He,π−)=(3.86±0.77(stat.)±0.68(syst.))×10−5 in the 0-10% most central collisions, is consistent with the predictions from a statistical thermal model using the same temperature as for the light hadrons. The coalescence parameter B3 shows a dependence on the transverse momentum, similar to the B2 of deuterons and the B3 of 3He nuclei. The ratio of yields S3 = 3ΛH/(3He ×Λ/p) was measured to be S3 = 0.60 ± 0.13 (stat.) ± 0.21 (syst.) in 0-10% centrality events; this value is compared to different theoretical models. The measured S3 is fully compatible with thermal model predictions. The measured 3ΛH lifetime, τ=181+54−39(stat.)±33(syst.) ps is compatible within 1σ with the world average value.