A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, Pósa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's theorem by proving that every graph with deg(u) + deg(v) ≥n for every uv ∉ E(G) contains a Hamiltonian cycle. Recently, Châu proved an Ore-type version of Pósa's conjecture for graphs on n≥n₀ vertices using the regularity—blow-up method; consequently the n₀ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n₀, we believe that our method of proof will be of independent interest.