I'd say: "My phone's memorycard went full in the middle of recording" (note, I have omitted "the"). The other sentence is ok. EDIT: Explanation: I omitted the definite article before "recording" because this word this word refers to the process of recording in general.
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TomerHello,My phone card filled up/ was full in the middle of recording.
"My phone memorycard went full in the middle of the recording"
and
"I recorded it with my phone, so the quality is not the best you can get"
Thank you