After nearly a century of zero-tolerance policies that treated gambling as a mortal threat, U.S. sports leagues are rapidly cutting deals with companies involved in sports betting. To varying degrees, the leagues are partnering — openly and in secret — with oddsmakers, betting prognosticators and data providers that make sports wagering possible in the digital age, according to interviews with a range of sports gambling officials and experts. The deals lay the groundwork for the NFL, the NBA, the NHL and Major League Baseball to profit if sports betting is legalized in the United States, as many experts predict. But...