SPRINGFIELD, Ill.—Illinois’s new governor delivered on a top campaign promise Tuesday by signing legislation legalizing small amounts of marijuana for recreational use, making the state the 11th to do so and the first to implement a comprehensive statewide cannabis marketplace designed by legislators.

Legalization in Illinois also means that nearly 800,000 people with criminal records for purchasing or possessing 30 grams of marijuana or less may have those records expunged, a provision minority lawmakers and interest groups...