We present cosmological constraints from the abundance of galaxy clusters selected via the thermal Sunyaev-Zel'dovich (SZ) effect in South Pole Telescope (SPT) data with a simultaneous mass calibration using weak gravitational lensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope (HST). The cluster sample is constructed from the combined SPT-SZ, SPTpol ECS, and SPTpol 500d surveys, and comprises 1,005 confirmed clusters in the redshift range 0.25–1.78 over a total sky area of 5200 deg2. We use DES Year 3 weak-lensing data for 688 clusters with redshifts z<0.95 and HST weak-lensing data for 39 clusters with 0.6<z<1.7. The weak-lensing measurements enable robust mass measurements of sample clusters and allow us to empirically constrain the SZ observable-mass relation without having to make strong assumptions about, e.g., the hydrodynamical state of the clusters. For a flat ΛCDM cosmology, and marginalizing over the sum of massive neutrinos, we measure Ωm=0.286±0.032, σ8=0.817±0.026, and the parameter combination σ8(Ωm/0.3)0.25=0.805±0.016. Our measurement of S8≡σ8Ωm/0.3=0.795±0.029 and the constraint from Planck CMB anisotropies (2018 TT, TE, EE+lowE) differ by 1.1σ. In combination with that Planck dataset, we place a 95% upper limit on the sum of neutrino masses ∑mν<0.18 eV. When additionally allowing the dark energy equation of state parameter w to vary, we obtain w=-1.45±0.31 from our cluster-based analysis. In combination with Planck data, we measure w=-1.34-0.15+0.22, or a 2.2σ difference with a cosmological constant. We use the cluster abundance to measure σ8 in five redshift bins between 0.25 and 1.8, and we find the results to be consistent with structure growth as predicted by the ΛCDM model fit to Planck primary CMB data.