Ms Grahame has supported the Margaret Kerr Unit since the public fundraising appeal was launched in September 2011.
She said: "I have been very impressed with the fundraising efforts of all those involved in this project and wrote to the team in November congratulating them.
"A palliative care unit has long been needed in the Borders as it meets the needs of patients and families facing progressive illness and bereavement. As we know up until now it has been the case that patients have to stay in Edinburgh or further afield, which can add to the distress for them and their families."
The Borders is the only mainland health authority area in Scotland which did not have a special in-patient facility offering palliative care and involving a multi-professional team. It is estimated around 200 people a year will use the Margaret Kerr Unit, including repeat admissions.
Ms Grahame added: "I have heard the new unit is an extremely tranquil and relaxing place to be and I am hoping to visit later this month to see for myself the fruits of many months of hard work."