diff --git a/mod.js b/mod.js
index 64dbf48..f9e3ad6 100644
--- a/mod.js
+++ b/mod.js
@@ -85,7 +85,7 @@ export class Importer {
    * [dynamic `import()` statement](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Statements/import#dynamic_imports).
    * Except with the faked modules applied.
    * @template T
-   * @param {string | URL} url
+   * @param {string} url
    * @returns {Promise<T>}
    */
   async import(url) {
diff --git a/src/ImportResolver.js b/src/ImportResolver.js
index fe75873..9d076d4 100644
--- a/src/ImportResolver.js
+++ b/src/ImportResolver.js
@@ -240,16 +240,18 @@ export class ImportResolver {
    * Once every file has loaded and its import urls replaced with blobs,
    * the entry point is imported with a regular async import call.
    * @template T
-   * @param {string | URL} url
+   * @param {string} url
    * @returns {Promise<T>}
    */
   async import(url) {
     this.#hasMadeImportCall = true;
-    if (typeof url === "string") {
-      url = new URL(url, this.#importMeta);
-    }
+    const newUrl = resolveModuleSpecifier(
+      this.#parsedImportMap,
+      new URL(this.#importMeta),
+      url,
+    );
     await this.loadImportMap();
-    const collectedImport = this.createCollectedImport(url.href);
+    const collectedImport = this.createCollectedImport(newUrl.href);
     let module;
     try {
       module = await import(await collectedImport.getBlobUrl());
diff --git a/test/integration/importMap.test.js b/test/integration/importMap.test.js
index fed5a18..c896237 100644
--- a/test/integration/importMap.test.js
+++ b/test/integration/importMap.test.js
@@ -29,7 +29,7 @@ Deno.test({
         },
       });
 
-      const module = await importer.import("main.js");
+      const module = await importer.import("./main.js");
       assertEquals(module.foo, "foo");
     } finally {
       await cleanup();
@@ -83,7 +83,7 @@ Deno.test({
       const importer = new Importer(basePath);
       importer.setImportMap("./importmap.json");
 
-      const module = await importer.import("main.js");
+      const module = await importer.import("./main.js");
       assertEquals(module.foo, "foo");
     } finally {
       await cleanup();
@@ -114,7 +114,7 @@ Deno.test({
         },
       });
 
-      const module = await importer.import("main.js");
+      const module = await importer.import("./main.js");
       assertEquals(module.foo, "foo");
     } finally {
       await cleanup();
@@ -146,7 +146,7 @@ Deno.test({
         },
       });
 
-      const module = await importer.import("main.js");
+      const module = await importer.import("./main.js");
       assertEquals(module.foo, "foo");
     } finally {
       await cleanup();
@@ -156,7 +156,6 @@ Deno.test({
 
 Deno.test({
   name: "importing a bare specifier directly without an import map",
-  ignore: true,
   async fn() {
     const { cleanup, basePath } = await simpleReplacementDir();
 
@@ -191,7 +190,7 @@ Deno.test({
       const importer = new Importer(basePath);
       await assertRejects(
         async () => {
-          await importer.import("main.js");
+          await importer.import("./main.js");
         },
         TypeError,
         `Relative import path "bare" not prefixed with / or ./ or ../`,
@@ -204,7 +203,6 @@ Deno.test({
 
 Deno.test({
   name: "importing a bare specifier directly that is not in the import map",
-  ignore: true,
   async fn() {
     const { cleanup, basePath } = await simpleReplacementDir();
 
@@ -249,7 +247,7 @@ Deno.test({
       });
       await assertRejects(
         async () => {
-          await importer.import("main.js");
+          await importer.import("./main.js");
         },
         TypeError,
         `Relative import path "bare" not prefixed with / or ./ or ../`,
@@ -272,7 +270,7 @@ Deno.test({
       const fullImportPath = new URL(importMapFileName, basePath);
       await assertRejects(
         async () => {
-          await importer.import("main.js");
+          await importer.import("./main.js");
         },
         TypeError,
         `Failed install import map from "${fullImportPath}". A network error occurred while fetching the module.`,
@@ -300,7 +298,7 @@ Deno.test({
       const fullImportPath = new URL(importMapFileName, basePath);
       await assertRejects(
         async () => {
-          await importer.import("main.js");
+          await importer.import("./main.js");
         },
         TypeError,
         `Failed install import map from "${fullImportPath}". The resource did not respond with an ok status code (404).`,