fix(route53): allow records with a weight of 0 (#29595)

### Issue # (if applicable)

Closes #29556 .

### Reason for this change

When creating an ARecord object in Route53 with props.weight set to 0 an Error is thrown.

### Description of changes

Change checks for `weight` to use `weight === undefined` instead of `!weight`

### Description of how you validated changes

- Added a unit test with `weight: 0`
- Updated integ tests to include a record with `weight: 0`

### Checklist
- [x] My code adheres to the [CONTRIBUTING GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and [DESIGN GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

packages/@aws-cdk-testing/framework-integ/test/aws-route53/test/integ.weighted-record.js.snapshot/weighted-record.template.json

@@ -53,6 +53,22 @@
     "Type": "A",
     "Weight": 50
    }
+  },
+  "WeightedRecord328406D0A": {
+   "Type": "AWS::Route53::RecordSet",
+   "Properties": {
+    "HostedZoneId": {
+     "Ref": "HostedZoneDB99F866"
+    },
+    "Name": "www.cdk.dev.",
+    "ResourceRecords": [
+     "4.5.6.7"
+    ],
+    "SetIdentifier": "WEIGHT_0_ID_weightedrecordWeightedRecord37AA25819",
+    "TTL": "10",
+    "Type": "A",
+    "Weight": 0
+   }
   }
  },
  "Parameters": {

packages/@aws-cdk-testing/framework-integ/test/aws-route53/test/integ.weighted-record.ts

@@ -15,6 +15,7 @@ class TestStack extends Stack {
       { target: '1.2.3.4', weight: 20 },
       { target: '2.3.4.5', weight: 30 },
       { target: '3.4.5.6', weight: 50 },
+      { target: '4.5.6.7', weight: 0 },
     ].forEach((data, index) => {
       new route53.ARecord(this, `WeightedRecord${index}`, {
         zone: hostedZone,

packages/aws-cdk-lib/aws-route53/lib/record-set.ts

@@ -302,7 +302,7 @@ export class RecordSet extends Resource implements IRecordSet {
     if (props.setIdentifier && (props.setIdentifier.length < 1 || props.setIdentifier.length > 128)) {
       throw new Error(`setIdentifier must be between 1 and 128 characters long, got: ${props.setIdentifier.length}`);
     }
-    if (props.setIdentifier && !props.weight && !props.geoLocation && !props.region && !props.multiValueAnswer) {
+    if (props.setIdentifier && props.weight === undefined && !props.geoLocation && !props.region && !props.multiValueAnswer) {
       throw new Error('setIdentifier can only be specified for non-simple routing policies');
     }
     if (props.multiValueAnswer && props.target.aliasTarget) {
@@ -405,7 +405,7 @@ export class RecordSet extends Resource implements IRecordSet {
       return identifier;
     }
 
-    if (this.weight) {
+    if (this.weight !== undefined) {
       const idPrefix = `WEIGHT_${this.weight}_ID_`;
       return this.createIdentifier(idPrefix);
     }

packages/aws-cdk-lib/aws-route53/test/record-set.test.ts

@@ -1133,6 +1133,39 @@ describe('record set', () => {
     });
   });
 
+  test('with weight of 0', () => {
+    // GIVEN
+    const stack = new Stack();
+
+    const zone = new route53.HostedZone(stack, 'HostedZone', {
+      zoneName: 'myzone',
+    });
+
+    // WHEN
+    new route53.RecordSet(stack, 'RecordSet', {
+      zone,
+      recordName: 'www',
+      recordType: route53.RecordType.CNAME,
+      target: route53.RecordTarget.fromValues('zzz'),
+      weight: 0,
+    });
+
+    // THEN
+    Template.fromStack(stack).hasResourceProperties('AWS::Route53::RecordSet', {
+      Name: 'www.myzone.',
+      Type: 'CNAME',
+      HostedZoneId: {
+        Ref: 'HostedZoneDB99F866',
+      },
+      ResourceRecords: [
+        'zzz',
+      ],
+      TTL: '1800',
+      Weight: 0,
+      SetIdentifier: 'WEIGHT_0_ID_RecordSet',
+    });
+  });
+
   test.each([
     [-1],
     [256],