A cross section of the process <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mo>±</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>∓</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> has been measured for the first time using a data sample of 185.4 pb$^{−1}$ collected with the CMD-3 detector at the VEPP-2000 <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></math> collider. With the <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> decay detection, 373±20 and 514±28 signal events have been selected with six and five reconstructed tracks, respectively, in the center-of-mass energy range 1.6–2.0 GeV. The total systematic uncertainty of the cross section is about 15%. A study of the production dynamics allows us to extract a contribution from the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>1285</mn><mo>)</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> intermediate state and to measure the corresponding cross section. The intermediate states with the <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>1420</mn><mo>)</mo></math> and <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>1510</mn><mo>)</mo></math> resonances have been observed.