The 33-year-old singer shared a touching photo of her and Polly on her Twitter page on Monday, with the message: "Heartbroken. Rest in peace my darling friend."
She shared a second picture on Instagram with the caption: "Heartbroken - sleep now with the angels."
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Katherine has previously spoken about her friendship with Polly, who she first met back in 2001 when they were both struggling singers.
Four years later, Polly was diagnosed with cervical cancer, and had to break the news over the phone to Katherine, who was on tour at the time.
"She didn't look or sound ill - it was just unbelievable," Katherine told You Magazine.
"And of course, the word cancer frightens the life out of me because I had watched my father die of lung cancer when I was 15.
"But he had been nearly 70," she added. "And Polly was just 24. She was too young; she had too much to live for."
Polly endured surgery, chemotherapy and radiotherapy, and for three years lived cancer-free. But, in 2010 she found a lump in her neck, which was revealed to be a secondary tumour.
"This time her oncologist told her that although she could have more treatment, the cancer was ultimately incurable." Katherine said.
"She called me from the hospital with the news. Neither of us cried. We were too stunned."
recently announced her engagement to American film director Andrew Levitas, and was no doubt hoping to have her best friend by her side on her special day.
Katherine and Andrew, 36, met while the were both working in New York and started dating in October. They publicly confirmed their romance in March, when Andrew joined the singer as she collected her OBE from Prince Charles at Buckingham Palace.
Friends have said the couple are likely to tie the knot in 2015 with a small ceremony in Katherine's native Wales followed by a second ceremony in the US.