We report results on an elastic cross section measurement in proton--proton collisions at a center-of-mass energy $\sqrt{s}=510$ GeV, obtained with the Roman Pot setup of the STAR experiment at the Relativistic Heavy Ion Collider (RHIC).The elastic differential cross section is measured in the four-momentum transfer squared range $0.23 \leq -t \leq 0.67$ GeV$^2$. This is the only measurement of the proton-proton elastic cross section in this $t$ range for collision energies above the Intersecting Storage Rings (ISR) and below the Large Hadron Collider (LHC) colliders.We find that a constant slope $B$ does not fit the data in the aforementioned $t$ range, and we obtain a much better fit using a second-order polynomial for $B(t)$. This is the first measurement below the LHC energies for which the non-constant behavior $B(t)$ is observed. The $t$ dependence of $B$ is determined using six subintervals of $t$ in the STAR measured $t$ range, and is in good agreement with the phenomenological models. The measured elastic differential cross section $\mathrm{d}\sigma/\mathrm{dt}$ agrees well with the results obtained at $\sqrt{s} = 540$ GeV for proton--antiproton collisions by the UA4 experiment. We also determine that the integrated elastic cross section within the STAR $t$-range is $\sigma^\mathrm{fid}_\mathrm{el} = 462.1 \pm 0.9 (\mathrm{stat.}) \pm 1.1 (\mathrm {syst.}) \pm 11.6 (\mathrm {scale})$ $\mu\mathrm{b}$.