Elastic electron-proton scattering (e-p) and the spectroscopy of hydrogen atoms are the two methods traditionally used to determine the proton charge radius, r(p). In 2010, a new method using muonic hydrogen atoms(1) found a substantial discrepancy compared with previous results(2), which became known as the 'proton radius puzzle'. Despite experimental and theoretical efforts, the puzzle remains unresolved. In fact, there is a discrepancy between the two most recent spectroscopic measurements conducted on ordinary hydrogen(3,4). Here we report on the proton charge radius experiment at Jefferson Laboratory (PRad), a high-precision e-p experiment that was established after the discrepancy was identified. We used a magnetic-spectrometer-free method along with a windowless hydrogen gas target, which overcame several limitations of previous e-p experiments and enabled measurements at very small forward-scattering angles. Our result, r(p) = 0.831 +/- 0.007(stat) +/- 0.012(syst) femtometres, is smaller than the most recent high-precision e-p measurement(5) and 2.7 standard deviations smaller than the average of all e-p experimental results(6). The smaller r(p) we have now measured supports the value found by two previous muonic hydrogen experiments(1,7). In addition, our finding agrees with the revised value (announced in 2019) for the Rydberg constant(8)-one of the most accurately evaluated fundamental constants in physics.