We present accretion-disk structure measurements from continuum lags in the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project. Lags are measured using the \texttt{JAVELIN} software from the first-year SDSS-RM \(g\) and \(i\) photometry, resulting in well-defined lags for 95 quasars, 33 of which have lag SNR \(>\) 2\(\sigma\). We also estimate lags using the \texttt{ICCF} software and find consistent results, though with larger uncertainties. Accretion-disk structure is fit using a Markov Chain Monte Carlo approach, parameterizing the measured continuum lags as a function of disk size normalization, wavelength, black hole mass, and luminosity. In contrast with previous observations, our best-fit disk sizes and color profiles are consistent (within 1.5~\(\sigma\)) with the \citet{SS73} analytic solution. We also find that more massive quasars have larger accretion disks, similarly consistent with the analytic accretion-disk model. The data are inconclusive on a correlation between disk size and continuum luminosity, with results that are consistent with both no correlation and with the \citet{SS73} expectation. The continuum lag fits have a large excess dispersion, indicating that our measured lag errors are underestimated and/or our best-fit model may be missing the effects of orientation, spin, and/or radiative efficiency. We demonstrate that fitting disk parameters using only the highest-SNR lag measurements biases best-fit disk sizes to be larger than the disk sizes recovered using a Bayesian approach on the full sample of well-defined lags.