* Firms in southern countries see biggest rate falls
* Continuing rate differentials remain headache for ECB
* ECB blames weak demand for slump in private sector loans
FRANKFURT, Feb 4 (Reuters) - The interest firms had to pay on loans ticked down in the euro zone in December, with southern European countries seeing the biggest falls, data showed on Tuesday, in a boost to the currency bloc's crisis-hit states.
The European Central Bank can take some solace from the development as evidence that its efforts to repair the so-called "fragmentation" of credit markets across the euro zone is having some success.
At the same time, differences in interest rates remain large. Firms in periphery states often face rates at least twice as high as their northern peers as banks in stressed countries face higher fund-raising costs and demand higher loan margins.
Funding costs for euro zone companies began to diverge as the financial crisis laid bare country-specific risks, prompting investors to demand premiums in some member states.
More than five years on, that divergence persists despite extraordinary measures implemented by the ECB. The gap is also a headache for the ECB, which sets policy for the euro zone average. However, the difference has begun to gradually narrow.
The ECB's measure of cross-country variation showed that corporate loan interest rates were at their closest in December since January 2012. The same measure showed that mortgage loan variation was the smallest since December 2011.
In December, corporate lending rates fell in most stressed countries, with Italy and Cyprus being the exceptions. In the periphery, funding costs varied between 3.35 percent in Spain and 5.79 percent in Cyprus. In Greece, they fell to a 3-1/2 year low of 5.41 percent.
Among the economically strong euro zone core countries, national average interest rates on new corporate loans varied between 1.99 percent in Luxembourg and 2.55 percent in Germany.
For the euro zone as a whole, corporate borrowing costs on new loans fell by 4 basis points to 2.94 percent, the biggest monthly drop since August, ECB data showed.
Household borrowing costs fell by 3 basis points to 3.05 percent, the first drop since June.
However, the difference between loan costs to large firms and smaller ones grew wider, with borrowing costs for loans with maturities longer than 1 year and of more than 1 million euros ticking down while those between 250,000 and 1 million euros remained unchanged.
Despite cheaper loans, lending to the private sector has slumped, with annual declines the largest in stressed southern European countries, including Spain and Italy.
ECB policymakers have said that weak lending is mainly due to a lack of demand in the 18-nation euro zone, with firms unwilling to invest when the economic outlook remains cloudy.
However, the central bank has lately started to accentuate the positive and, while stressing that the recovery is still tenuous, has said financial market improvements seem to be working their way into the real economy.
In the coming months, banks expect loan demand to increase across the board and have also said they are done tightening lending rules, the ECB's bank lending survey showed last week.
The ECB has high hopes for ongoing bank health checks, expecting that confidence in the banking system will return as markets form a clearer picture of its well-being and that this will lead to smaller differences in bank funding costs. (Editing by Gareth Jones)