Designated smoking rooms for patients and staff will close and smoking in the hospital grounds will be outlawed, as the final stage of the ban comes into effect.
It means long-term patients who have to stay on the hospital site at all times will now have nowhere to smoke, and staff will have to leave during breaks if they want a cigarette.
The first stage of the ban was rolled out last April when smokers were forbidden from lighting up within a five-yard radius of all hospital entrances.
No smoking signs were put up outside, but a hospital spokesman said many visitors have "blatantly ignored" them and continued to smoke.
Then in January this year, the ban was extended across the hospital except for the three designated smoking rooms and within the grounds.
From next Friday, visitors caught smoking inside the buildings or anywhere within the grounds will be asked to stop and staff could face disciplinary action.
When the ban was first proposed in February last year, it sparked concern from nurses who feared it would create a fire risk and lead to more pressure on staff who smoke to de-stress.
This week a spokesman said the hospital will be offering staff and patients specialist help if they want to use the ban as an opportunity to quit smoking.
The spokesman said: "We are relying on goodwill of visitors. People are generally very sensible and we believe they will understand we are doing this with the best of intentions."