SEOUL, May 6 (Reuters) - South Korea's worst outbreak of bird flu has spread to the capital with agriculture ministry officials saying on Tuesday four dead birds raised at a Seoul district government office had tested positive.

South Korea has confirmed 24 cases of the deadly H5N1 strain in poultry in about a month. It has killed more than 5 million chickens and ducks but that has not stopped the virus from spreading at its fastest rate since the country reported its first case in 2003.

The four birds were among 57 chickens, ducks and turkeys raised in pens at the district office in the eastern part of Seoul. The ministry said it would take about a day to confirm if the dead birds had contracted the H5N1 strain.

The city had culled all the birds at the site and was limiting access to a nearby zoo and livestock market, the ministry said.

Most of the outbreaks have been concentrated in the southwest of the country.

South Korea had to kill 5.29 million birds during the first outbreak between late 2003 and early 2004. The second outbreak in 2006-2007 saw about half that number culled.

On Monday, North Korea said it was stepping up measures to prevent the outbreak from being carried over its borders by migratory flocks.

The last known bird flu outbreak in impoverished North Korea was in 2005. (Reporting by Lee Jiyeon and Jon Herskovitz; Editing by Nick Macfie)

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