How can I apply a border only inside a table?

I am trying to figure out how to add border only inside the table. When I do:

table {
    border: 0;
}
table td, table th {
    border: 1px solid black;
}

The border is around the whole table and also between table cells. What I want to achieve is to have border only inside the table around table cells (without outer border around the table).

Here is markup I’m using for tables (even though I think that is not important):

<table>
    <tr>
        <th>Heading 1</th>
        <th>Heading 2</th>
    </tr>
    <tr>
        <td>Cell (1,1)</td>
        <td>Cell (1,2)</td>
    </tr>
    <tr>
        <td>Cell (2,1)</td>
        <td>Cell (2,2)</td>
    </tr>
    <tr>
        <td>Cell (3,1)</td>
        <td>Cell (3,2)</td>
    </tr>
</table>

And here are some basic styles I apply to most of my tables:

table {
    border-collapse: collapse;
    border-spacing: 0;
}