We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state ρ = v | ψ − ⟩ ⟨ ψ − | + ( 1 − v ) 1 4 via a local hidden variable (LHV) model, where | ψ − ⟩ denotes the singlet state. We show analytically that these correlations are local for v = 999 × 689 × 10 − 6 cos 2 ⁡ ( π / 50 ) ≃ 0.6829 . In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant K G ( 3 ) ≤ 1 / v ≃ 1.4644 . We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v ≃ 0.4553 . The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.