Precision Measurements of the Nucleon Strange Form Factors at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>∼</mml:mo><mml:mn>0.1</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:msup><mml:mi>GeV</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>

We report new measurements of the parity-violating asymmetry A(PV) in elastic scattering of 3 GeV electrons off hydrogen and 4He targets with <theta(lab)> approximately 6.0 degrees . The 4He result is A(PV)=(+6.40+/-0.23(stat)+/-0.12(syst))x10(-6). The hydrogen result is A(PV)=(-1.58+/-0.12(stat)+/-0.04(syst))x10(-6). These results significantly improve constraints on the electric and magnetic strange form factors G(E)(s) and G(M)(s). We extract G(E)(s)=0.002+/-0.014+/-0.007 at <Q(2)>=0.077 GeV2, and G(E)(s)+0.09G(M)(s)=0.007+/-0.011+/-0.006 at <Q(2)>=0.109 GeV2, providing new limits on the role of strange quarks in the nucleon charge and magnetization distributions.