We investigated the magnetoelectric properties of a new laminate composite material based on y + 140°-cut congruent lithium niobate piezoelectric plates with an antiparallel polarized “head-to-head” bidomain structure and metglas used as a magnetostrictive layer. A series of bidomain lithium niobate crystals were prepared by annealing under conditions of Li2O outdiffusion from LiNbO3 with a resultant growth of an inversion domain. The measured quasi-static magnetoelectric coupling coefficient achieved |αE31| = 1.9 V·(cm Oe)−1. At a bending resonance frequency of 6862 Hz, we found a giant |αE31| value up to 1704 V·(cm Oe)−1. Furthermore, the equivalent magnetic noise spectral density of the investigated composite material was only 92 fT/Hz1/2, a record value for such a low operation frequency. The magnetic-field detection limit of the laminated composite was found to be as low as 200 fT in direct measurements without any additional shielding from external noises.