Near-Earth asteroid (NEA) 1566 Icarus ($a=1.08\,\mathrm{au}$, e = 0.83, $i=22\buildrel{\circ}\over{.} 8$) made a close approach to Earth in 2015 June at 22 lunar distances (LD). Its detection during the 1968 approach (16 LD) was the first in the history of asteroid radar astronomy. A subsequent approach in 1996 (40 LD) did not yield radar images. We describe analyses of our 2015 radar observations of Icarus obtained at the Arecibo Observatory and the DSS-14 antenna at Goldstone. These data show that the asteroid is a moderately flattened spheroid with an equivalent diameter of 1.44 km with 18% uncertainties, resolving long-standing questions about the asteroid size. We also solve for Icarus's spin-axis orientation ($\lambda =270^\circ \pm 10^\circ ,\beta =-81^\circ \pm 10^\circ $), which is not consistent with the estimates based on the 1968 light-curve observations. Icarus has a strongly specular scattering behavior, among the highest ever measured in asteroid radar observations, and a radar albedo of ~2%, among the lowest ever measured in asteroid radar observations. The low cross section suggests a high-porosity surface, presumably related to Icarus's cratering, spin, and thermal histories. Finally, we present the first use of our orbit-determination software for the generation of observational ephemerides, and we demonstrate its ability to determine subtle perturbations on NEA orbits by measuring Icarus's orbit-averaged drift in semimajor axis ($(-4.62\pm 0.48)\times {10}^{-4}$ au My−1, or ~60 m per revolution). Our Yarkovsky rate measurement resolves a discrepancy between two published rates that did not include the 2015 radar astrometry.