We introduce a new iterative amalgamated free product construction of II 1 factors, and use it to construct a separable II 1 factor which does not have property Gamma and is not elementarily equivalent to the free group factor L ( F n ) , for any 2 ≤ n ≤ ∞ . This provides the first explicit example of two non-elementarily equivalent II 1 factors without property Gamma. Moreover, our construction also provides the first explicit example of a II 1 factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.