The British model and former sports star, who had son Ethan with Australia's Got Talent judge Minogue during their four-year relationship, said that it has been a very "tough" time.
Smith wrote on his Twitter page: "I would like to thank you all for your very kind words and support at this tough time... It really means a lot.
"Dannii and I have a deep mutual love and respect for each other and although things have not worked out we have a gorgeous son together that we both love and implore everyone to respect our privacy at this challenging time."
Reports surfaced yesterday that Minogue and Smith's relationship deteriorated after he allegedly gave her a new passport holder for Christmas, seen as a hint that he was unhappy about the amount of time she spent away from home.
Other sources have claimed that Minogue became "colder" and was "a nightmare" after the birth of their child.
Minogue confirmed their separation on Twitter, writing: "It brings me great sadness to tell you that Kris and I have separated. We still care for each other and ask for privacy at this difficult time."
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