Prosecutor Andrew Edis told the Old Bailey that messages from William left on Kate's mobile phone, including one in which he called her "babykins", had been discovered at the house of the paper's former royal editor and a private eye working for the tabloid in 2006.
William and Kate met as students at St Andrew's University in Scotland in 2001 and married in a spectacular ceremony in April, 2011, watched by up to two billion people globally.
The court also heard extracts of transcripts of a message William left for his younger brother Harry in which he put on a high voice and pretended to be Harry's then girlfriend Chelsy Davy and called him "ginger", referring to his hair colour.
Former editors Andy Coulson and Rebekah Brooks are currently on trial with five others accused of a variety of offences which they all deny, including conspiracy to illegally intercept voicemails from mobiles.
While it was known that royal aides had previously been targeted by the paper, it was the first time it had been disclosed that any royal family members themselves were victims.
In August 2006 the royal editor Clive Goodman and private investigator Glenn Mulcaire were arrested and later charged with hacking the telephones of royal aides by accessing voicemail messages.
In January 2007, both Goodman and Mulcaire admitted the charges and were sentenced to four and six months imprisonment respectively.
Mulcaire has now pleaded guilty to further phone-hacking charges while three senior journalists from the tabloid have also admitted conspiracy to tap mobile messages.