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The relationship between galaxy star formation rates (SFR) and stellar masses ($M_\ast$) is re-examined using a mass-selected sample of $\sim$62,000 star-forming galaxies at $z \le 1.3$ in the COSMOS 2-deg$^2$ field. Using new far-infrared photometry from $Herschel$-PACS and SPIRE and $Spitzer$-MIPS 24 $\mu$m, along with derived infrared luminosities from the NRK method based on galaxies' locations in the restframe color-color diagram $(NUV - r)$ vs. $(r - K)$, we are able to more accurately determine total SFRs for our complete sample. At all redshifts, the relationship between median $SFR$ and $M_\ast$ follows a power-law at low stellar masses, and flattens to nearly constant SFR at high stellar masses. We describe a new parameterization that provides the best fit to the main sequence and characterizes the low mass power-law slope, turnover mass, and overall scaling. The turnover in the main sequence occurs at a characteristic mass of about $M_{0} \sim 10^{10} M_{\odot}$ at all redshifts. The low mass power-law slope ranges from 0.9-1.3 and the overall scaling rises in SFR as a function of $(1+z)^{4.12 \pm 0.10}$. A broken power-law fit below and above the turnover mass gives relationships of $SFR \propto M_{*}^{0.88 \pm 0.06}$ below the turnover mass and $SFR \propto M_{*}^{0.27 \pm 0.04}$ above the turnover mass. Galaxies more massive than $M_\ast \gtrsim 10^{10}\ M_{\rm \odot}$ have on average, a much lower specific star formation rate (sSFR) than would be expected by simply extrapolating the traditional linear fit to the main sequence found for less massive galaxies.