In this paper, we continue to build support for the proposal to use gamma-ray bursts (GRBs) as standard candles in constructing the Hubble Diagram at redshifts beyond the current reach of Type Ia supernova observations. We confirm that correlations among certain spectral and lightcurve features can indeed be used as luminosity indicators, and demonstrate from the most up-to-date GRB sample appropriate for this work that the $Λ$CDM model optimized with these data is characterized by parameter values consistent with those in the concordance model. Specifically, we find that $(Ω_m,Ω_Λ)≈ (0.25_{-0.06}^{+0.05}, 0.75_{-0.05}^{+0.06})$, which are consistent, to within $1σ$, with $(0.29,0.71)$ obtained from the 9-yr WMAP data. We also carry out a comparative analysis between $Λ$CDM and the $R_{\rm h}=ct$ Universe and find that the optimal $Λ$CDM model fits the GRB Hubble Diagram with a reduced $χ^2_{\rm dof}≈ 2.26$, whereas the fit using $R_{\rm h}=ct$ results in a $χ^2_{\rm dof}≈ 2.14$. In both cases, about 20% of the events lie at least $2σ$ away from the best-fit curves, suggesting that either some contamination by non-standard GRB luminosities is unavoidable, or that the errors and intrinsic scatter associated with the data are being underestimated. With these optimized fits, we use three statistical tools---the Akaike Information Criterion (AIC), the Kullback Information Criterion (KIC), and the Bayes Information Criterion (BIC)---to show that, based on the GRB Hubble Diagram, the likelihood of $R_{\rm h}=ct$ being closer to the correct model is $∼ 85-96%$, compared to $∼ 4-15%$ for $Λ$CDM. ; Comment: 39 pages, 7 figures, 2 tables, accepted for publication in ApJ