Conjugated diene rubber composition for tire use and tire

A conjugated diene rubber composition for tires, comprising:

• (I) a high-cis conjugated diene polymer rubber, and
• (II) a second polymer rubber that is different from the high-cis conjugated diene polymer rubber (I);
• wherein the amounts of (I) and (II) are 2% to 100% by weight and 98% to 0% by weight, respectively, based on the total weight of the polymer rubbers;
• and
• wherein said high-cis conjugated diene polymer rubber is a polymer of 1,3-butadiene monomer or a copolymer of 1,3-butadiene monomer with other conjugated diene monomer or with a monomer other than the conjugated diene monomer;
• the content of units having a cis-bond in the total units derived from the 1,3-butadiene monomer and the conjugated diene monomer is at least 80%;
• the weight average molecular weight (Mw) of said high-cis conjugated diene polymer rubber is in the range of 20,000 to 10,000,000;
• the ratio (Mw/Mn) of weight average molecular weight (Mw) to number average molecular weight (Mn) in said high-cis conjugated diene polymer rubber is not larger than 1.9;
• and said ratio (Mw/Mn) and said weight average molecular weight (Mw) satisfy the following formula (1):
• log(Mw/Mn)<0.162×log(Mw)−0.682 (1).