You can certainly do that with some custom CSS. Here is an example that you can start with:
.header-image-link img {
display: none;
}
.header-image-link {
display: block;
background: url(YOUR_IMAGE_URL) no-repeat;
width: 390px;
height: 273px;
margin: 0 auto;
}
Replace YOUR_IMAGE_URL with a link to your image which is sized to 390 x 273 pixels. If you have any trouble, you should include a link to the WordPress.com blog you're working with, a link to your header image, and the code you've tried already in a reply.