Automatically close database in sqlite3 of Python

In Python sqlite3, the context manager

with sqlite3.connect(database)

would automatically commit or rollback transactions. However, it does not close the connection nor the cursor. So every time I have to write something like this:

with sqlite3.connect(database) as conn:
    cur = conn.cursor()
    cur.execute(an_sql_statement)
    cur.executemany(more_sql_statements)

cur.close()
conn.close()

which becomes quite repetitive.

I want to be able to do something like this:

with AutoCloseDB(database) as cur:
    cur.execute(an_sql_statement)
    cur.executemany(more_sql_statements)

and the above context manager would automatically commit or rollback transactions, close the cursor and finally close the database altogether upon exiting.

So I came up with the following context manager:

import sqlite3

class AutoCloseDB:
    """A context manager that automatically closes the cursor and the database.
    Return a cursor object upon entering.
    """

    def __init__(self, database):
        self.conn = sqlite3.connect(database)

    def __enter__(self):
        self.conn = self.conn.__enter__()
        self.cur = self.conn.cursor()
        return self.cur

    def __exit__(self, *exc_info):
        result = self.conn.__exit__(*exc_info)
        self.cur.close()
        self.conn.close()
        return result

Is there any problem in the above code? Is there any better solution?