We report on a measurement of the parity violating asymmetry in the elastic scattering of polarized electrons off unpolarized protons with the A4 apparatus at MAMI in Mainz at a four momentum transfer value of Q(2) = 0.108 (GeV/c)(2) and at a forward electron scattering angle of 30 degrees <theta(e)< 40 degrees. The measured asymmetry is A(LR)((e) over right arrow )p = [-.36 +/- 0.29(stat) +/- 0.13(syst)] x 10(-6). The expectation from the standard model assuming no strangeness contribution to the vector current is A(0)=(-2.06 +/- 0.14) x 10(-6). We have improved the statistical accuracy by a factor of 3 as compared to our previous measurements at a higher Q(2). We have extracted the strangeness contribution to the electromagnetic form factors from our data to be G(E)(s) + 0.106G(M)(s) = 0.071 +/- 0.036 at Q(2) = 0.108 (GeV/c)(2). We again find the value for G(E)(s) + 0.106G(M)(s) to be positive, this time at an improved significance level of two sigma.