You can do:
#!/bin/bash
# Link file descriptor #6 with stdout/stderr
exec 6>&1 2>&1
exec 1> record.log 2>&1
echo begin redirecting
# restore and close file descriptor #6
exec 1>&6 2>&1 6>&-
# from the script above, 'begin redirecting' will be redirected to record.log
# then we want to disable the redirecting and restore the STDOUT&STDERR
echo return to STDOUT and STDERR