The unusual time slot for the fifth-round match is an attempt to raise awareness of mental health.
In England, matches in the FA Cup third round last month all kicked off 60 seconds later than usual to encourage fans to ‘Take A Minute’ to think about looking after their mental health, as part of the Heads Up campaign.
Hamilton have followed suit.
The Lanarkshire club said in a tweet: "We kick off one minute later to urge you to ‘take a minute’ to talk to your friends and fellow supporters, to help us breakdown the stigma and encourage conversation around mental health."
The Hamilton v Rangers tie at the Foys Stadium will be broacast live by Premier Sports.