Hi, I have achieved it in 1.6.1.20 PS instance by following modifications (with autoupdate):
File: \classes\Cart.php (but you should use override here) -> added following lines:
3167: $total_weight = $this->getTotalWeight();
3256: 'total_weight' => $total_weight,
File: \themes\yourtheme\js\cart-summary.js -> added following line:
958: $('#total_weight').html(json.total_weight);
File: \themes\yourtheme\shopping-cart.tpl -> added following lines in proper place of your template shopping-cart.tpl file:
<tr class="cart_total_weight"> <td colspan="{$col_span_subtotal}" class="text-right">{l s='Total weight'}</td> <td colspan="4" class="price"><span id="total_weight">{$cart->getTotalWeight()}</span> {Configuration::get('PS_WEIGHT_UNIT')}</td> </tr>