Murray Ballantyne sent messages to the teenager in February 2022 asking her about her underwear and inviting her to his Elm Row home to have sexual intercourse.
The messages were seen by the girl's older sister who informed the police.
Prosecutor India McLean said Ballantyne met the girl who was in a group of teenagers and gave her his mobile number in case she required anything.
The fiscal said he would agree to meet her to buy her cigarettes and he continued to message her inviting her to have sex and made comments to her about having an erection.
The girl's sister was concerned about the behaviour and contacted the police and told the 13-year-old girl she was doing so.
The teenager then told her older sister about other sexualised messages and phone calls.
Ballantyne was cautioned and charged and made no reply.
He pleaded guilty to charges of communicating indecently with a child and purchasing cigarettes for an under age person.
Sheriff Kevin McCarron deferred sentence until December 11 and ordered that Ballantyne's name be placed on the Sex Offenders Register