A vulcanizable rubber mixture and its use for rubber products, which need good grip in wet or icy conditions, in particular vehicle tires and vehicle tire treads, but also technical rubber products like belts on conveyor systems, technical hoses, or vibration dampers and impact dampers. The vulcanizable rubber mixture includes from 10 to 100 phr of a block copolymer functionalized at least at one chain end (terminally) for binding to fillers, at least one filler and, optionally, additives. The block copolymer contains a statistically or microsequentially polymerized copolymer comprising a conjugated diene and an aromatic vinyl compound, and has, polymerized onto at least one end of the main chain, a terminally functionalized block of different structure to the main chain, which structure is homopolymeric or copolymeric, from 5 to 250 monomer units long, and composed of at least one of the mers of the main chain copolymer.