Since I broke my arm, I can’t swim for a month.

"Since" has two meanings:

1. In the intervening period between (the time mentioned) and the time under consideration, typically the present.
"she has suffered from depression since she was sixteen"

2. For the reason that; because.
"delegates were delighted, since better protection of rhino reserves will help protect other rare species"

That introduces ambiguity, especially with the expression of time in the second part, and the fact that American English is much more likely to use the simple past "I broke my arm" in situations where British English uses the present perfect "I have broken my arm". . Do you mean:

Because I broke my arm, I have been unable to swim for one month.

-or-

I have broken my arm, and will be unable to swim for the next month.

...?

The sentence is awkward in American usage, as well. One might say: Since I broke my arm, I am not allowed to swim for a month.

It seems good to me.

Thanks to all of you.