Vulcanizable rubber mixture which comprises at least one block copolymer functionalized at least at one chain end (terminally) for binding to fillers, at least one filler and, if appropriate, additives, characterized in that the mixture comprises from 10 to 100 phr of a statistically or microsequentially polymerized copolymer comprising a conjugated diene and an aromatic vinyl compound, and which has, polymerized onto at least one end of said main chain copolymer, a terminally functionalized block of different structure to the main chain, which structure is internally homopolymeric, from 20 to 180 monomer units long, and composed of one of the mers of the main chain copolymer, with the provisio that the glass transition temperatures on the one hand of the main chain and on the other hand of the terminal block or block polymer are different, whereas identical or different functional groups as listed below are present at at least one chain end or at both chain ends of the block copolymer consisting of main chain polymer and terminal block, the degree of functionalization, based on the block copolymer, being from 30% to 100%: -COOH, -COX, where X = halogen, -SH, -CSSH, -NCO, amino, silyl, silanol or siloxane groups inclusive of, in each case attached to the polymer chain with or without a spacer, polysiloxane groups and siloxane and polysiloxane groups comprising amino groups, in which the amino groups can be represented by the following formulae: -A-N(R 1) 2, -A-NHR 1, -A-NH 2, the silyl, silanol and siloxane groups can be represented by the following formulae: -A-SiH 2, -A-Si(R 1) 2, -A-SiH 2, -A-SiR 1 2, -A-Si 3,-A-Si(OR 1) 3, -A-(SiR 1 R 2 O) x -R 3, -A-Si(R 3) 3, -A-Si(R 3 /X) 3, where X = halogen and A = spacer group, and the siloxane groups comprising amino groups can be represented via the following formula: -A 1 -Si(A 2 -N(H/R 1) 2) n (OR 1) m (R 3) 3-(n+m) in each case where R 1 and R 2 are identical or different, being namely alkoxy or alkyl, branched or straight-chain, cycloalkyl, aryl, alkylaryl, aralkyl, or vinyl, in each case having from 1 to 20 carbon atoms, and mononuclear aryl, x = a whole number from 1 to 1500, R 3 is -H, or -alkyl, branched or straight-chain, or cycloalkyl, in each case having from 1 to 20 carbon atoms, or mononuclear aryl, and A 1, A 2 = a C 0 -C 12 organic spacer chain, branched or unbranched, preferably C 0 -C 12 -alkyl, -allyl or -vinyl, whereas microsequential means that one or both comonomers is present at least partially in the form of microblocks, whereas the microblocks encompass from 2 to 10, more preferably from 3 to 10, and in particular from 3 to 6, connected units of a comonomer. in each case where R 1 and R 2 are identical or different, being namely alkoxy or alkyl, branched or straight-chain, cycloalkyl, aryl, alkylaryl, aralkyl, or vinyl, in each case having from 1 to 20 carbon atoms, and mononuclear aryl, x = a whole number from 1 to 1500, R 3 is -H, or -alkyl, branched or straight-chain, or cycloalkyl, in each case having from 1 to 20 carbon atoms, or mononuclear aryl, and A 1, A 2 = a C 0 -C 12 organic spacer chain, branched or unbranched, preferably C 0 -C 12 -alkyl, -allyl or -vinyl, whereas microsequential means that one or both comonomers is present at least partially in the form of microblocks, whereas the microblocks encompass from 2 to 10, more preferably from 3 to 10, and in particular from 3 to 6, connected units of a comonomer.