Corporate jet Captain reported braking abruptly short of the runway they were cleared to cross because they observed an aircraft on the takeoff roll on that runway.

Narrative

Landed Runway XX ZZZ at XA25 local time on Day 0 and exited left on taxiway 1. Contacted Ground Control and advised parking at FBO. Was instructed to taxi via taxiways 1; 2; 3; 4; cross runways XYR and XYL. Upon turning left on taxiway 3 and approaching the hold short line for runway XYR; I observed an aircraft on the runway with its lights on. After several seconds I determined the aircraft was taking off on runway XYR which I was about to cross. I braked immediately and stopped the aircraft short of the runway. I advised Ground Control that we were holding short of the runway that they had cleared us to cross because an aircraft was taking off on that runway. The conflicting aircraft was already airborne when it passed abeam our location. I asked for the Controller to clarify our taxi clearance; and taxied to the FBO without further incident. After arriving at the FBO; I requested a phone number to debrief the event with a Tower supervisor. I called the supervisor at XB09 and advised him of the incident. He advised that he would investigate and call me back. I received a call back at approximately XB40 and was told that my account of the incident was correct; and that the Controller had cleared me to cross both runways and then subsequently cleared the other aircraft to takeoff on Runway XYR.Possible action to prevent a recurrence of this type of event is to not clear an aircraft to taxi across an active runway when it is more than a mile away from the point of crossing.