I think you will find that extra minute without too much bother. Springle has quite rightly given you advice which should see your aerobice fitness levels increase. This is important, and 5 weeks gives you some chance to improve this.
However, equally important is that you will also learn how to pace your run so that you can run the whole thing, without slowing very much, or stopping. Similarly you will find that you will learn how to hurt your body more, and keep going when you are stopping at the moment.
I halped my daughter prepare for her 800m race at school sports day. She only had 2 weeks, so no time to really impact her actual fitness. Yet by practicing doing 800m several times, she ended up knocking 52s off her time simply because she could pace it properly and was prepared to tough it out.
I suggest the plan Springle said, but do your next 1.5m slower, at a pace you know you can maintain for the whole thing. Then each week, do that run slightly faster.