The STAR Collaboration reports measurements of the longitudinal double-spin asymmetry, A(LL), for neutral pions produced at forward directions in polarized proton-proton collisions, at a center-of-mass energy of 510 GeV. Results are given for transverse momenta in the range 2 < p(T) < 10 GeV= c within two regions of pseudorapidity that span 2.65 < eta < 3.9. These results are sensitive to the polarized gluon parton distribution function, Delta g(chi) down to the region of Bjorken x similar to 10(-3). The asymmetries observed are less than similar to 5 x 10(-3) in magnitude and will help constrain the contribution to the spin of the proton from polarized gluons at low x, when combined with other measurements as part of a global analysis.