We derive cosmological constraints using a galaxy cluster sample selected from the 2500~deg$^2$ SPT-SZ survey. The sample spans the redshift range $0.25< z<1.75$ and contains 343 clusters with SZ detection significance $xi>5$. The sample is supplemented with optical weak gravitational lensing measurements of 32 clusters with $0.29<z<1.13$ (from Magellan and HST) and X-ray measurements of 89 clusters with $0.25<z<1.75$ (from Chandra). We rely on minimal modeling assumptions: i) weak lensing provides an accurate means of measuring halo masses, ii) the mean SZ and X-ray observables are related to the true halo mass through power-law relations in mass and dimensionless Hubble parameter $E(z)$ with a-priori unknown parameters, iii) there is (correlated, lognormal) intrinsic scatter and measurement noise relating these observables to their mean relations. We simultaneously fit for these astrophysical modeling parameters and for cosmology. Assuming a flat $ uLambda$CDM model, in which the sum of neutrino masses is a free parameter, we measure $Omega_mathrm{m}=0.276pm0.047$, $sigma_8=0.781pm0.037$, and $sigma_8(Omega_mathrm{m}/0.3)^{0.2}=0.766pm0.025$. The redshift evolution of the X-ray $Y_mathrm{X}$-mass and $M_mathrm{gas}$-mass relations are both consistent with self-similar evolution to within $1sigma$. The mass-slope of the $Y_mathrm{X}$-mass relation shows a $2.3sigma$ deviation from self-similarity. Similarly, the mass-slope of the $M_mathrm{gas}$-mass relation is steeper than self-similarity at the $2.5sigma$ level. In a $ u w$CDM cosmology, we measure the dark energy equation of state parameter $w=-1.55pm0.41$ from the cluster data. We perform a measurement of the growth of structure since redshift $zsim1.7$ and find no evidence for tension with the prediction from General Relativity. We provide updated redshift and mass estimates for the SPT sample. (abridged)
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