A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (6b0342da383b45a78757a0a" title="Click to view the MathML source">v1,e1,v2,…,vn,en) of distinct vertices v1,…,vn and distinct hyperedges e1,…,en such that {v1,vn}⊆en and {vi,vi+1}⊆ei for every 26bfb" title="Click to view the MathML source">i∈[n−1]. We prove a Dirac-type theorem for Hamilton Berge cycles in random r -uniform hypergraphs by showing that for every integer r≥3 there exists k=k(r) such that for every b0516f561d2a8f71837524db0c67" title="Click to view the MathML source">γ>0 and asymptotically almost surely every spanning subhypergraph H⊆H(r)(n,p) with minimum vertex degree contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on p is optimal up to possibly the logarithmic factor. As a corollary this gives a new upper bound on the threshold of H(r)(n,p) with respect to Berge Hamiltonicity.