Measurements of the elastic scattering cross section of 3He and 4He are important in order to improve constraints on theoretical models of 4He(3He,g)7Be, a key reaction in Big Bang nucleosynthesis and solar neutrino production. The astrophysical S-factor for this reaction is a significant source of uncertainty in the standard solar-model prediction of the 7Be and 8B solar neutrino fluxes. The elastic scattering measurements reported in the literature do not extend to low energies and lack proper uncertainty quantification. A new measurement of the 4He(3He,3He)4He reaction has been made at center-of-mass energies Ec.m. = 0.38-3.13 MeV using the Scattering of Nuclei in Inverse Kinematics (SONIK) scattering chamber: a windowless, extended gas target surrounded by an array of 30 collimated silicon charged particle detectors situated at TRIUMF. This is the first elastic scattering measurement of 3He+4He made below 500 keV and it has greater angular range and better precision than previous measurements. The elastic scattering data were analyzed using both R-matrix and Halo Effective Field Theory (Halo EFT) frameworks, and values of the s-wave scattering length and effective range were extracted. The resulting improvement in knowledge of the s-wave effective-range function at low energies will reduce the overall uncertainty in S34 at solar energies.