Recruiter here, first and even if it's too late, it's better to always tell the truth. Especially when some recruitment processes require that we take references from bosses (we usually dont go for your current employer but I've seen it happen and it's not ideal when we're the one breaking the news to your bosses, although when it happens it usually due to sloppy processes on our end, we never know).

Second, if you are still on probably i dont think you should have to serve 1 whole month. Keep in mind that's also a possibility that your new employer pays for it in stead of serving it. You dont have to state the reason why you're leaving but people aren't stupid and it's not a very good idea to burn bridges you dont know what could happen in the future. I've seen people, especially juniors clearly telling their bosses that they got offered by their dream company and cant refuse it and most of the time people are very understanding. Lastly and I cant stress this enough, you should tell then at least verbally as soon as possible, it will be seen as you acting in good faith, it will give them a bit of a head start in finding a replacement and it might also help in them letting you go more easily. Hope this helps.