ABSTRACT We implement the first blind analysis of cluster abundance data to derive cosmological constraints from the abundance and weak lensing signal of redMaPPer clusters in the Sloan Digital Sky Survey (SDSS). We simultaneously fit for cosmological parameters and the richness–mass relation of the clusters. For a flat Λ cold dark matter cosmological model with massive neutrinos, we find $S_8 \equiv \sigma _{8}(\Omega _\mathrm{ m}/0.3)^{0.5}=0.79^{+0.05}_{-0.04}$. This value is both consistent and competitive with that derived from cluster catalogues selected in different wavelengths. Our result is also consistent with the combined probes analyses by the Dark Energy Survey (DES), the Kilo-Degree Survey (KiDS), and with the cosmic microwave background (CMB) anisotropies as measured by Planck. We demonstrate that the cosmological posteriors are robust against variation of the richness–mass relation model and to systematics associated with the calibration of the selection function. In combination with baryon acoustic oscillation data and big bang nucleosynthesis data (Cooke et al.), we constrain the Hubble rate to be h = 0.66 ± 0.02, independent of the CMB. Future work aimed at improving our understanding of the scatter of the richness–mass relation has the potential to significantly improve the precision of our cosmological posteriors. The methods described in this work were developed for use in the forthcoming analysis of cluster abundances in the DES. Our SDSS analysis constitutes the first part of a staged-unblinding analysis of the full DES data set.